Skills Beta

Math you need: the one-page sheet

Functions and Their Graphs

A function sends each input to one output. Read it from a formula (substitute with parentheses), a graph (heights, filled and open dots) or a table. Know the domain, write lines in point-slope form, compose functions inside first, move graphs with f(x − h) + k , and find vertical asymptotes where a denominator is 0.

A function gives one output for each input. Substitute with parentheses: if $f(x) = x^2 - 3x$ , then $f(a + 1)$ $= (a + 1)^2 - 3(a + 1)$ .
The domain is every input the rule accepts; the range is every output it produces. Rule out zero denominators and square roots of negatives.
On a graph, a filled dot is on the graph and an open circle is not. $f(a)$ is the height of the graph at $x = a$ .
A line through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$ (point-slope form). Slope is rise over run.
$f(g(x))$ means $g$ first, then $f$ . Order matters: usually $f(g(x)) \ne g(f(x))$ .
A rational function has a vertical asymptote where the denominator is 0 and the numerator is not.

f(3) names one number: the height of the graph above x = 3 those inputs are left out of the domain f(g(x)) means: do g first, then f moves the whole graph h units right and k units up

function: A rule that assigns exactly one output to each input. <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(3)</annotation></semantics></math> is the output when the input is 3.
domain: The set of inputs a function accepts. The range is the set of outputs it produces.
graph of a function: All points <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, f(x))</annotation></semantics></math> for <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the domain. The height of the graph above <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math> is <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math>; a filled dot is on the graph, an open circle is not.
slope: Rise over run: for a line through <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, y_1)</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_2, y_2)</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math><wbr><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mfrac><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">= \frac{y_2 - y_1}{x_2 - x_1}</annotation></semantics></math>.
point-slope form: The point-slope form of a line with slope <math class="calc-m"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> through <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, y_1)</annotation></semantics></math>: <math class="calc-m"><semantics><mrow><mi>y</mi><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><mi>m</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y - y_1 = m(x - x_1)</annotation></semantics></math>. Slope-intercept form is <math class="calc-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y = mx + b</annotation></semantics></math>.
piecewise function: A function defined by different formulas on different parts of its domain, such as <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x) = x^2</annotation></semantics></math> for <math class="calc-m"><semantics><mrow><mi>x</mi><mo><</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">x < 2</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f(x) = 3x - 1</annotation></semantics></math> for <math class="calc-m"><semantics><mrow><mi>x</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">x \ge 2</annotation></semantics></math>.
composite function: A composite function feeds one output into another function: <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(g(x))</annotation></semantics></math> applies <math class="calc-m"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> first, then <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.
graph transformation: A change that moves or reshapes a graph: <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f(x - h) + k</annotation></semantics></math> shifts it <math class="calc-m"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> right and <math class="calc-m"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> up; <math class="calc-m"><semantics><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-f(x)</annotation></semantics></math> flips it over the <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>-axis; <math class="calc-m"><semantics><mrow><mi>a</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a f(x)</annotation></semantics></math> stretches it vertically.
rational function: A quotient of two polynomials, such as <math class="calc-m"><semantics><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{2x + 1}{x - 4}</annotation></semantics></math>. A polynomial is a sum of terms <math class="calc-m"><semantics><mrow><mi>a</mi><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a x^n</annotation></semantics></math> with whole-number powers <math class="calc-m"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.
asymptote: A line the graph approaches. A rational function has a vertical asymptote <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math> where its denominator is 0 and its numerator is not; a horizontal asymptote is a height the outputs settle toward as <math class="calc-m"><semantics><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math> grows large.

Exponent Rules and Rewriting Powers

Exponent rules let you write every root and fraction as a power: 1/(x^n) = x^(−n) and n√(x^m) = x^(m/n) . Add exponents to multiply, subtract to divide, multiply for a power of a power, and remember an exponent applies only to the base it touches and never splits over a sum.

Multiply powers of the same base: add exponents. Divide: subtract. Power of a power: multiply.
$x^{-n} = \frac{1}{x^n}$ . A negative exponent means reciprocal, never a negative number.
$x^{m/n} = \left(\sqrt[n]{x}\right)^m$ . Take the root first to keep numbers small: $8^{2/3} = 2^2 = 4$ .
An exponent applies only to what it touches: $3x^{-2} = \frac{3}{x^2}$ , but $(3x)^{-2} = \frac{1}{9x^2}$ .
Exponents do not split over sums: $(x + 3)^{-1} \ne x^{-1} + 3^{-1}$ and $\sqrt{x^2 + 9} \ne x + 3$ .

so x^m⋅x^n = x^(m + n) , and (x^m)/(x^n) = x^(m − n) a negative exponent is a reciprocal: x^(−n) = 1/(x^n) a fractional exponent is a root: x^(1/2) = √x and x^(m/n) = (n√x)^m expressions like 3/x² + 2√x become 3x⁻² + 2x^(1/2) , ready for calculus

exponent rules: The laws for powers of one base: <math class="calc-m"><semantics><mrow><msup><mi>x</mi><mi>m</mi></msup><msup><mi>x</mi><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">x^m x^n = x^{m+n}</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mfrac><msup><mi>x</mi><mi>m</mi></msup><msup><mi>x</mi><mi>n</mi></msup></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{x^m}{x^n} = x^{m-n}</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>m</mi></msup><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(x^m)^n = x^{mn}</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^0 = 1</annotation></semantics></math>. A negative exponent is a reciprocal (<math class="calc-m"><semantics><mrow><msup><mi>x</mi><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><msup><mi>x</mi><mi>n</mi></msup></mfrac></mrow><annotation encoding="application/x-tex">x^{-n} = \frac{1}{x^n}</annotation></semantics></math>); a fractional exponent is a root (<math class="calc-m"><semantics><mrow><msup><mi>x</mi><mrow><mi>m</mi><mo>/</mo><mi>n</mi></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mroot><mi>x</mi><mi>n</mi></mroot><msup><mo stretchy="false">)</mo><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">x^{m/n} = (\sqrt[n]{x})^m</annotation></semantics></math>).

Algebra for Limits: Factoring and Rationalizing

Limits that give zero over zero are settled with precalculus algebra: factor and cancel a common factor, multiply by a conjugate to clear a square root, or combine a complex fraction into one. Cancel factors, never terms, and expand powers in full. The simplified expression agrees with the original everywhere except the troublesome point.

Factoring toolkit: common factor; difference of squares $a^2 - b^2 = (a - b)(a + b)$ ; trinomials (with the $ac$ method when the leading coefficient is not 1); cubes $a^3 \pm b^3$ $= (a \pm b)(a^2 \mp ab + b^2)$ .
Cancel factors, never terms: $\frac{x(x - 4)}{(x - 4)(x + 4)}$ $= \frac{x}{x + 4}$ , but $\frac{x^2 + 4}{x + 2}$ does not simplify.
Rationalizing: multiply top and bottom by the conjugate ( $\sqrt{A} - B$ and $\sqrt{A} + B$ form a pair). Their product is $A - B^2$ . Leave the other factor unmultiplied.
A complex fraction becomes one fraction with a common denominator; then divide.
Expand powers fully: $(a + h)^2 = a^2 + 2ah + h^2$ . The middle term is the one that matters.

factor top and bottom so that factor is visible gives a simpler expression equal to the original for every x ≠ a multiplying by the conjugate √x + 2 turns it into x − 4 , which can cancel a common denominator turns the complex fraction into one fraction you can simplify

factoring: Writing an expression as a product, such as <math class="calc-m"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^2 - 9 = (x - 3)(x + 3)</annotation></semantics></math> (a difference of squares). A factored form shows which factors can cancel.
rationalizing: Removing a square root from part of a fraction by multiplying top and bottom by the conjugate: <math class="calc-m"><semantics><mrow><msqrt><mi>A</mi></msqrt><mo>−</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\sqrt{A} - B</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><msqrt><mi>A</mi></msqrt><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\sqrt{A} + B</annotation></semantics></math> are conjugates, and <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><msqrt><mi>A</mi></msqrt><mo>−</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msqrt><mi>A</mi></msqrt><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sqrt{A} - B)(\sqrt{A} + B)</annotation></semantics></math><wbr><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>A</mi><mo>−</mo><msup><mi>B</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">= A - B^2</annotation></semantics></math>.
complex fraction: A fraction whose top or bottom contains fractions, such as <math class="calc-m"><semantics><mrow><mfrac><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{\frac{1}{x + 2} - \frac{1}{2}}{x}</annotation></semantics></math>. Combine the inner fractions over a common denominator to simplify it.

Trigonometry and the Unit Circle

Radians measure angles by arc length on the unit circle, where the point at angle θ is (cos θ, sin θ) . Exact values come from the first-quadrant special angles plus the quadrant signs. Tangent, secant, cosecant and cotangent are ratios of sine and cosine, and sin² x + cos² x = 1 links them. Keep calculators in radian mode.

Radians: $\pi$ radians $= 180^{\circ}$ . To convert degrees to radians, multiply by $\frac{\pi}{180}$ . Calculus always uses radians, including on the calculator.
On the unit circle, the point at angle $\theta$ is $(\cos\theta, \sin\theta)$ : cosine is the $x$ -coordinate, sine the $y$ -coordinate.
Know the first-quadrant values at $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ : $\frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$ . Other quadrants use a reference angle and a sign.
$\sec = \frac{1}{\cos}$ , $\csc = \frac{1}{\sin}$ , $\cot = \frac{\cos}{\sin}$ . $\sin^2 x$ means $(\sin x)^2$ .
Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ ; also $1 + \tan^2 x = \sec^2 x$ and $\sin 2x = 2\sin x\cos x$ .

radians: a half turn is π , a full turn 2π the point reached is (cos θ, sin θ) , and tan θ = (sin θ)/(cos θ) sin² θ + cos² θ = 1 , the Pythagorean identity gives the exact value at any special angle

radian: A measure of angle: the length of the arc the angle cuts on a circle of radius 1. A full turn is <math class="calc-m"><semantics><mrow><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">2\pi</annotation></semantics></math> radians, so <math class="calc-m"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> radians <math class="calc-m"><semantics><mrow><mo>=</mo><msup><mn>180</mn><mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">= 180^{\circ}</annotation></semantics></math>.
unit circle: The circle of radius 1 centered at the origin, <math class="calc-m"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 + y^2 = 1</annotation></semantics></math>. The point at angle <math class="calc-m"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> (measured counterclockwise from <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 0)</annotation></semantics></math>) is <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>,</mo><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cos\theta, \sin\theta)</annotation></semantics></math>.
sine: Sine and cosine give the coordinates of the point at angle <math class="calc-m"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> on the unit circle: <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>,</mo><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cos\theta, \sin\theta)</annotation></semantics></math>. The others are ratios: <math class="calc-m"><semantics><mrow><mi>tan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow><mrow><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\tan\theta = \frac{\sin\theta}{\cos\theta}</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>sec</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\sec\theta = \frac{1}{\cos\theta}</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>csc</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\csc\theta = \frac{1}{\sin\theta}</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>cot</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow><mrow><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\cot\theta = \frac{\cos\theta}{\sin\theta}</annotation></semantics></math>.
Pythagorean identity: An equation true for every angle. The Pythagorean identity <math class="calc-m"><semantics><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sin^2\theta + \cos^2\theta = 1</annotation></semantics></math> comes from the unit circle; dividing by <math class="calc-m"><semantics><mrow><msup><mi>cos</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">\cos^2\theta</annotation></semantics></math> gives <math class="calc-m"><semantics><mrow><msup><mi>tan</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi><mo>+</mo><mn>1</mn><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">\tan^2\theta + 1 = \sec^2\theta</annotation></semantics></math>.

Exponentials and Logarithms

e^x is the exponential function calculus uses; it is always positive and passes through (0, 1) . ln x , defined for x > 0 , undoes it: e^(ln x) = x and ln (e^x) = x . Logs turn products into sums, quotients into differences and powers into multipliers, which is how you solve for a variable in an exponent.

$e^x > 0$ for every $x$ ; $e^0 = 1$ . The graph rises steeply to the right and approaches the $x$ -axis to the left.
$\ln x$ is defined only for $x > 0$ ; $\ln 1 = 0$ , $\ln e = 1$ . Its graph is the graph of $e^x$ reflected across $y = x$ .
Log rules: $\ln(ab) = \ln a + \ln b$ , $\ln\frac{a}{b} = \ln a - \ln b$ , $\ln(a^p) = p\ln a$ . There is no rule for $\ln(a + b)$ .
$e$ and $\ln$ cancel: $e^{\ln x} = x$ ( $x > 0$ ) and $\ln(e^x) = x$ . Use them to solve equations.
$A e^{kt}$ grows when $k > 0$ and decays when $k < 0$ ; the time to double is $\frac{\ln 2}{k}$ .

it models growth by a fixed percentage; the base e ≈ 2.718 is the one calculus prefers e^(ln x) = x for x > 0 and ln (e^x) = x for every x give the log rules: ln (ab) = ln a + ln b , ln a/b = ln a − ln b , ln a^p = pln a take ln of both sides to bring it down: 2x = ln 9

exponential function: A function <math class="calc-m"><semantics><mrow><msup><mi>b</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">b^x</annotation></semantics></math> with a fixed positive base <math class="calc-m"><semantics><mrow><mi>b</mi><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b \ne 1</annotation></semantics></math>; it changes by the same factor over equal steps in <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. The natural exponential <math class="calc-m"><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">e^x</annotation></semantics></math> uses <math class="calc-m"><semantics><mrow><mi>e</mi><mo>≈</mo><mn>2.71828</mn></mrow><annotation encoding="application/x-tex">e \approx 2.71828</annotation></semantics></math>. <math class="calc-m"><semantics><mrow><mi>A</mi><msup><mi>e</mi><mrow><mi>k</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A e^{kt}</annotation></semantics></math> shows exponential growth when <math class="calc-m"><semantics><mrow><mi>k</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k > 0</annotation></semantics></math> and exponential decay when <math class="calc-m"><semantics><mrow><mi>k</mi><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k < 0</annotation></semantics></math>.
natural logarithm: <math class="calc-m"><semantics><mrow><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\ln x</annotation></semantics></math> is the power of <math class="calc-m"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> that gives <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>: <math class="calc-m"><semantics><mrow><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\ln x = y</annotation></semantics></math> means <math class="calc-m"><semantics><mrow><msup><mi>e</mi><mi>y</mi></msup><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">e^y = x</annotation></semantics></math>. It is defined for <math class="calc-m"><semantics><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x > 0</annotation></semantics></math>. Log rules: <math class="calc-m"><semantics><mrow><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>a</mi><mo>+</mo><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\ln(ab) = \ln a + \ln b</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>=</mo><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>a</mi><mo>−</mo><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\ln\frac{a}{b} = \ln a - \ln b</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><msup><mi>a</mi><mi>p</mi></msup><mo>=</mo><mi>p</mi><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\ln a^p = p\ln a</annotation></semantics></math>.

Inverse Functions

An inverse function runs a one-to-one function backward: f⁻¹(b) = a exactly when f(a) = b . Its graph is the reflection of f across y = x , so points swap coordinates and slopes become reciprocals. Find a formula by solving x = f(y) for y . The inverse trig functions use restricted ranges, and f⁻¹ never means 1/f .

$f^{-1}(b) = a$ means $f(a) = b$ . So $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ on the right domains.
$f^{-1}(x)$ is not $\frac{1}{f(x)}$ . The $-1$ means "undo", not "reciprocal".
Only a one-to-one function (each output from one input; passes the horizontal line test) has an inverse. Restrict the domain if needed.
To find a formula: write $x = f(y)$ and solve for $y$ .
$\arcsin x$ and $\arctan x$ return angles in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ ; $\arccos x$ returns angles in $[0, \pi]$ .

the inverse function f⁻¹ sends b back to a : f⁻¹(b) = a exactly when f(a) = b swaps the coordinates of every point, reflecting the graph across y = x slopes at matching points are reciprocals their inverses use restricted ranges: arcsine and arctangent give angles from −π/2 to π/2, arccosine from 0 to π

inverse function: For a one-to-one function <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, the function <math class="calc-m"><semantics><mrow><msup><mi>f</mi><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> that undoes it: <math class="calc-m"><semantics><mrow><msup><mi>f</mi><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">f^{-1}(b) = a</annotation></semantics></math> exactly when <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f(a) = b</annotation></semantics></math>. Its graph is the reflection of the graph of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> across <math class="calc-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y = x</annotation></semantics></math>. A function is one-to-one when no two inputs share an output.
inverse trigonometric function: <math class="calc-m"><semantics><mrow><mi>arcsin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\arcsin x</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>arccos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\arccos x</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>arctan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\arctan x</annotation></semantics></math> (also written <math class="calc-m"><semantics><mrow><msup><mi>sin</mi><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mn>1</mn></mrow></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin^{-1} x</annotation></semantics></math>, ...) give the angle with a given sine, cosine or tangent, from a restricted range: <math class="calc-m"><semantics><mrow><mrow><mo fence="true" stretchy="true">[</mo><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>,</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow><mo fence="true" stretchy="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]</annotation></semantics></math> for arcsine, <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0, \pi]</annotation></semantics></math> for arccosine, <math class="calc-m"><semantics><mrow><mrow><mo fence="true" stretchy="true">(</mo><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>,</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow><mo fence="true" stretchy="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)</annotation></semantics></math> for arctangent.

Geometry for Calculus

Calculus word problems start with a geometric equation. Right angles call for the Pythagorean theorem; nested triangles call for similar triangles and a ratio; shapes call for their area or volume formulas. Label changing quantities with letters, fixed ones with numbers, and use a ratio to reduce a formula to one variable.

Pythagorean theorem: in a right triangle with legs $a$ , $b$ and hypotenuse $c$ , $a^2 + b^2 = c^2$ . The hypotenuse is opposite the right angle.
Similar triangles have equal angles, so their sides are in proportion: $\frac{a}{b} = \frac{A}{B}$ . Look for a small triangle inside a big one.
Know the area and volume formulas: circle $\pi r^2$ , semicircle $\frac{1}{2}\pi r^2$ , trapezoid $\frac{b_1 + b_2}{2}h$ , cylinder $\pi r^2 h$ , cone $\frac{1}{3}\pi r^2 h$ , sphere $\frac{4}{3}\pi r^3$ .
Draw the picture, label what changes with letters and what is fixed with numbers, then write one equation.

the Pythagorean theorem links the sides: x² + y² = z² similar triangles: corresponding sides are in the same ratio, such as r/h = 4/10 a formula like V = 1/3 πr²h becomes a function of one variable, V = 4π/75 h³

similar triangles: Triangles with the same three angles. Their corresponding sides are in the same ratio, so <math class="calc-m"><semantics><mrow><mfrac><mi>r</mi><mi>h</mi></mfrac><mo>=</mo><mfrac><mi>R</mi><mi>H</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{r}{h} = \frac{R}{H}</annotation></semantics></math> for matching sides.
Pythagorean theorem: In a right triangle with legs <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> and hypotenuse <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>: <math class="calc-m"><semantics><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">a^2 + b^2 = c^2</annotation></semantics></math>.
volume of a cone: Volume of a cone <math class="calc-m"><semantics><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></mrow><annotation encoding="application/x-tex">\frac{1}{3}\pi r^2 h</annotation></semantics></math>, of a cylinder <math class="calc-m"><semantics><mrow><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></mrow><annotation encoding="application/x-tex">\pi r^2 h</annotation></semantics></math>, of a sphere <math class="calc-m"><semantics><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\frac{4}{3}\pi r^3</annotation></semantics></math>; area of a trapezoid <math class="calc-m"><semantics><mrow><mfrac><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac><mi>h</mi></mrow><annotation encoding="application/x-tex">\frac{b_1 + b_2}{2}h</annotation></semantics></math>, of a semicircle <math class="calc-m"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\frac{1}{2}\pi r^2</annotation></semantics></math>.