Functions, Graphs and Notation — AP® Precalculus

Part 1 · Hook

Why this matters

Press B4 on a vending machine and you always get the same snack. That one-button, one-snack rule is what makes a function: every input gives exactly one output. Precalculus is the study of functions, so this page sets up the words and notation the whole course uses.

Part 2 · See it

See it first

The graph of f(x) = square root of (x + 4), minus 1, starts at the closed point (−4, −1), crosses the x-axis at the zero (−3, 0), crosses the y-axis at (0, 1) and passes through (5, 2). Its domain is x ≥ −4 and its range is y ≥ −1. — The graph of $f(x) = \sqrt{x + 4} - 1$ . Read inputs across and outputs up: the zero is $x = -3$ , the $y$ -intercept is $(0, 1)$ and $f(5) = 2$ . LevlPrep original diagram.

Part 3 · Step by step

How it works, step by step

Each input of a function has exactly one output $f(3) = 7$ names one point $(3, 7)$ on the graph of $f$
A formula divides by an expression or takes its square rootthe domain leaves out inputs that make a denominator $0$ or a square root's inside negative
An input $a$ gives the output $0$ $a$ is a zero of $f$ and the graph meets the $x$ -axis at $(a, 0)$
A piecewise-defined function is evaluatedfirst find the piece whose condition holds for the input, then use only that piece

Part 4 · Key ideas

Key ideas

Function: each input gives exactly one output. $f(3) = 7$ means the input $3$ gives the output $7$ .
Domain: the inputs a function accepts. Range: the outputs it produces.
Zero: an input with output $0$ , an $x$ -intercept of the graph. $y$ -intercept: the point $(0, f(0))$ .
To evaluate, replace every $x$ with the input in parentheses: $f(-2)$ , $f(x + 1)$ .

Part 5 · Misconception

A common mistake

The wrong idea: $f(x + 1)$ is the same as $f(x) + 1$ .

What actually happens: $f(x + 1)$ changes the input: replace every $x$ with $x + 1$ . $f(x) + 1$ changes the output: add $1$ after evaluating. For $f(x) = x^2$ they are $(x + 1)^2$ and $x^2 + 1$ , which differ.

Part 6 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. For $f(x) = x^2 - 3x + 1$ , what is $f(-2)$ ?

$11$
$3$
$-1$
$-9$

Show the answer

Put $-2$ in for every $x$ , inside parentheses: $f(-2)$ $= (-2)^2 - 3(-2)$ $+ 1$ $= 4$ $+ 6$ $+ 1$ $= 11$ .

Correct: $11$ : Right: $(-2)^2 - 3(-2) + 1$ $= 4 + 6 + 1$ $= 11$ .
$3$ : This squares only the $2$ , as if $(-2)^2$ were $-4$ . With parentheses, $(-2)^2 = 4$ .
$-1$ : This uses $-3(-2) = -6$ . A negative times a negative is positive: $-3(-2) = 6$ .
$-9$ : Two sign slips: $(-2)^2$ read as $-4$ and $-3(-2)$ read as $-6$ .

2. For $f(x) = x^2 - 3x + 1$ , write $f(x + 1)$ as an expression in $x$ .

Type an expression for $f(x + 1) =$ .

Show the answer

Replace each $x$ with $(x + 1)$ : $(x + 1)^2 - 3(x + 1)$ $+ 1$ $= x^2$ $+ 2x$ $+ 1 - 3x - 3$ $+ 1$ $= x^2 - x - 1$ .

Answer: $f(x + 1) =$ $(x + 1)^{2} - 3(x + 1) + 1$

3. What is the domain of $g(x)$ $= \frac{\sqrt{x - 2}}{x - 5}$ ?

$x \ge 2$ and $x \ne 5$
$x \ge 2$
$x \ne 5$
$x > 2$ and $x \ne 5$

Show the answer

Two rules: the expression under a square root cannot be negative ( $x \ge 2$ ), and a denominator cannot be $0$ ( $x \ne 5$ ). Both must hold.

Correct: $x \ge 2$ and $x \ne 5$ : Right: the square root needs $x - 2 \ge 0$ , and the denominator needs $x - 5 \ne 0$ .
$x \ge 2$ : This misses the denominator: at $x = 5$ it is $0$ , and division by $0$ is undefined.
$x \ne 5$ : This misses the square root: for $x < 2$ , $x - 2$ is negative and has no real square root.
$x > 2$ and $x \ne 5$ : $x = 2$ is allowed: $\sqrt{0} = 0$ , and $\frac{0}{-3} = 0$ is a real number.

Graph

The graph of $y = g(x)$

4. Use the graph to find $g(0)$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

The graph crosses the $y$ -axis at $(0, -1.5)$ , so $g(0) = -1.5$ : the $y$ -intercept.

Answer: -1.5

5. Which table does NOT define $y$ as a function of $x$ ?

$x$ : $1, 2, 2, 3$ and $y$ : $4, 5, 6, 7$
$x$ : $1, 2, 3, 4$ and $y$ : $5, 5, 5, 5$
$x$ : $-1, 0, 1, 2$ and $y$ : $1, 0, 1, 4$
$x$ : $0, 3, 6, 9$ and $y$ : $9, 6, 3, 0$

Show the answer

A function gives each input exactly one output. Look for a repeated input with different outputs; repeated outputs are fine.

Correct: $x$ : $1, 2, 2, 3$ and $y$ : $4, 5, 6, 7$ : Right: the input $2$ has two outputs, $5$ and $6$ , so this is not a function.
$x$ : $1, 2, 3, 4$ and $y$ : $5, 5, 5, 5$ : Every input has one output. Several inputs may share an output.
$x$ : $-1, 0, 1, 2$ and $y$ : $1, 0, 1, 4$ : Each input appears once, so each has exactly one output. $-1$ and $1$ sharing $1$ is allowed.
$x$ : $0, 3, 6, 9$ and $y$ : $9, 6, 3, 0$ : Each input appears once with one output: a function.

6. What is the zero of $h(x) = 2x + 10$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Set the output to $0$ : $2x + 10 = 0$ , $2x = -10$ , $x = -5$ . Check: $h(-5) = 0$ .

Answer: -5

Part 7 · Summary

Summary

A function gives each input one output, written

f(\text{input}) = \text{output}

. The domain is every allowed input (no zero denominators, no square roots of negatives); the range is every output. Zeros are inputs with output

0

; the

y

-intercept is

(0, f(0))

. Piecewise-defined functions use the piece whose condition the input meets.

Part 8 · Up next

What comes next

Part 9 · Connections

Connections

Next: Interval Notation and Inequalities

Why this matters

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graph of y=g(x)y = g(x)

Summary

What comes next

Connections

The graph of $y = g(x)$