Unit 4 · Topic 4.5

Implicitly Defined Functions

An implicitly defined function is an equation linking x and y ; its graph is every solution point.

Practice 2: Multiple Representations

Beyond the exam. The AP® Precalculus exam does not test this unit. Many classes teach it, and it prepares you for calculus.

Part 1 · Hook

Why this matters

The edge of a pond, the outline of a running track, the shape of a lens: many curves are easiest to describe with one equation that links x and y, not with a formula "y = ...". Such an equation still lets you find points, measure the curve and split it into pieces that are functions.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Solve $y^2 + y - 6 = 0$ .

$y = 2$ or $y = -3$
$y = -2$ or $y = 3$
$y = 6$ or $y = -1$
$y = 2$

Show the answer

$y^2 + y - 6$ $= (y - 2)(y + 3)$ $= 0$ , so $y = 2$ or $y = -3$ .

Correct: $y = 2$ or $y = -3$ :
$y = -2$ or $y = 3$ :
$y = 6$ or $y = -1$ :
$y = 2$ :

2. Does the graph of $x = y^2$ pass the vertical line test?

No: $x = 4$ gives $y = 2$ and $y = -2$
Yes: each $y$ gives one $x$
Yes: it is a parabola
No: it has no $y$ -intercept

Show the answer

The vertical line $x = 4$ meets the graph at $(4, 2)$ and $(4, -2)$ , two points, so $y$ is not a function of $x$ .

Correct: No: $x = 4$ gives $y = 2$ and $y = -2$ :
Yes: each $y$ gives one $x$ :
Yes: it is a parabola:
No: it has no $y$ -intercept:

Part 4 · See it

See it first

The circle x squared plus y squared = 25. Its upper half, y = the square root of 25 minus x squared, is drawn in green and its lower half, y = minus the square root of 25 minus x squared, in blue. A dashed vertical line at x = 3 meets the circle twice, at (3, 4) and (3, -4). — $x^2 + y^2 = 25$ is not the graph of a function of $x$ : the line $x = 3$ meets it twice. Its upper and lower halves are each the graph of a function. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

An equation links $x$ and $y$ without solving for $y$ its graph is every point $(x, y)$ that makes the equation true
Substitute a value of $x$ you get an equation in $y$ alone; each solution gives a point of the curve
Some $x$ gives two or more values of $y$ the curve is not the graph of a function of $x$ , but it can be split into pieces that are
For some $x$ the equation in $y$ has no real solutionthe curve has no points above or below that $x$ , which bounds how far left or right it reaches

Part 6 · Key ideas

Key ideas

An implicitly defined function is given by an equation in $x$ and $y$ , such as $x^2 + xy + y^2 = 7$ , instead of $y = f(x)$ . Its graph is the set of solutions.
To test a point, substitute both coordinates. To find points, substitute one coordinate and solve for the other.
If one $x$ gives two $y$ -values, the curve is not a function of $x$ . Solving for $y$ (sometimes with the quadratic formula) splits it into functions.
An implicit equation can define a function you cannot easily solve for, such as $y^3 + y = x$ .

Part 7 · Misconception

A common mistake

The wrong idea: An equation that is not written as $y = \ldots$ cannot define a function.

What actually happens: What matters is whether each $x$ gives one $y$ . $y^3 + y = x$ gives exactly one $y$ for each $x$ , so it defines $y$ as a function of $x$ even though the formula is hard to write.

Part 8 · Check yourself

Check yourself

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

Table

Points on the curve $y^3 + y = x$

Values of y and the matching x = y cubed + y
$y$	−2	−1	0	1	2
$x = y^3 + y$	−10	−2	0	2	10

1. Find $x$ for the point of the curve with $y = 3$ .

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

Show the answer

$x$ $= (3)^3 + 3$ $= 27 + 3$ $= 30$ .

Answer: 30

2. Which value of $y$ goes with $x = 10$ ?

$y = 2$
$y = 1010$
$y = 5$
$y = 2$ and $y = -2$

Show the answer

The table shows $y = 2$ gives $x = 2^3 + 2 = 10$ . No other $y$ works, since $y^3 + y$ increases as $y$ increases.

Correct: $y = 2$ : $8 + 2 = 10$ .
$y = 1010$ : This puts $10$ in for $y$ ; $10$ is the $x$ -value.
$y = 5$ : $5^3 + 5 = 130$ .
$y = 2$ and $y = -2$ : $y = -2$ gives $x = -10$ , not $10$ .

3. Does $y^3 + y = x$ define $y$ as a function of $x$ ?

Yes: $y^3 + y$ increases with $y$ , so each $x$ has one $y$
No: it is not written in the form $y = f(x)$ , so it is no function
No: $y$ is cubed, so $x = 2$ has three values of $y$
Yes: the table lists five different values of $y$

Show the answer

$y^3$ and $y$ both increase with $y$ , so their sum does too. Two different $y$ -values give two different $x$ -values, so each $x$ has one $y$ .

Correct: Yes: $y^3 + y$ increases with $y$ , so each $x$ has one $y$ : An increasing left side means no $x$ repeats.
No: it is not written in the form $y = f(x)$ , so it is no function: The form does not decide it; one $y$ per $x$ does.
No: $y$ is cubed, so $x = 2$ has three values of $y$ : $y^3 + y = 2$ has one real solution, $y = 1$ .
Yes: the table lists five different values of $y$ : Five rows cannot show what happens for every $x$ ; the reason is that $y^3 + y$ increases.

4. The average rate of change of $y$ with respect to $x$ is $\frac{1}{2}$ between $(-2, -1)$ and $(2, 1)$ and $\frac{1}{8}$ between $(2, 1)$ and $(10, 2)$ . What does this suggest?

The curve gets less steep as $x$ increases past $2$
The curve gets steeper as $x$ increases past $2$
$y$ decreases as $x$ increases past $2$
The curve is a straight line through the origin

Show the answer

Both rates are positive, so $y$ increases, but the second is smaller: each unit of $x$ adds less to $y$ . The curve flattens.

Correct: The curve gets less steep as $x$ increases past $2$ : A smaller positive slope means less steep.
The curve gets steeper as $x$ increases past $2$ : Steeper would need a larger slope; $\frac{1}{8} < \frac{1}{2}$ .
$y$ decreases as $x$ increases past $2$ : Positive rates mean $y$ increases with $x$ .
The curve is a straight line through the origin: A line has one slope; these differ.

5. What is an implicitly defined function?

An equation linking $x$ and $y$ , not solved for $y$
A function whose formula has no $x$ in it
A function given by a table of values and nothing else
A pair of formulas $x(t)$ and $y(t)$ for a curve

Show the answer

"Implicit" means the relationship is stated as an equation linking $x$ and $y$ ; the graph is every solution point.

Correct: An equation linking $x$ and $y$ , not solved for $y$ : An equation links $x$ and $y$ ; points are its solutions.
A function whose formula has no $x$ in it: A formula with no $x$ is a constant function, written explicitly.
A function given by a table of values and nothing else: A table gives values, not an equation.
A pair of formulas $x(t)$ and $y(t)$ for a curve: That describes a parametric function.

6. Which points of $x^2 + y^2 = 25$ have $x = 3$ ?

$(3, 4)$ and $(3, -4)$
$(3, 4)$ , the point above the axis
$(3, 16)$ and $(3, -16)$
$(3, 22)$ and $(3, -22)$

Show the answer

$9 + y^2 = 25$ , so $y^2 = 16$ and $y = 4$ or $y = -4$ .

Correct: $(3, 4)$ and $(3, -4)$ : Both square roots of $16$ .
$(3, 4)$ , the point above the axis: $y^2 = 16$ also has the root $y = -4$ .
$(3, 16)$ and $(3, -16)$ : $16$ is $y^2$ ; take the square root.
$(3, 22)$ and $(3, -22)$ : $25 - 3 = 22$ subtracts $3$ , not $3^2$ .

Part 9 · Summary

Summary

An implicitly defined function is an equation linking

x

and

y

; its graph is every solution point. Substitute a value to find points, count the

y

-values for one

x

to decide whether the curve is a function of

x

, and solve for

y

to split it into functions.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

Points on the curve y3+y=xy^3 + y = x

Summary

What comes next

Connections

Points on the curve $y^3 + y = x$