Reflecting Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Reflecting Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Reflecting a function mirrors its graph across the $x$ -axis ( $-f(x)$ ), $y$ -axis ( $f(-x)$ ), or the line $y = x$ (the inverse function).

$-f(x)$ flips over x-axis (upside down). $f(-x)$ flips over y-axis (mirror).

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Reflecting mirrors a graph: $-f(x)$ flips it over the $x$ -axis, $f(-x)$ flips it over the $y$ -axis.

Common stuck point: The procedure for reflecting functions is the easy part; the trap is swapping which negative flips which axis. Asking "Is the graph a mirror image of the parent caused by a negative sign (not a slide or a stretch)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the graph a mirror image of the parent caused by a negative sign (not a slide or a stretch)?

Worked Examples

Example 1

easy

Given

f(x)=x^3-2

, write the equations for (a) reflection over the

x

-axis and (b) reflection over the

y

-axis. Evaluate each at

x=2

Answer

(a)

g(x)=-x^3+2

g(2)=-6

; (b)

h(x)=-x^3-2

h(2)=-10

First step

(a) Reflection over

x

-axis: negate the output →

g(x)=-f(x)=-(x^3-2)=-x^3+2

g(2)=-8+2=-6

Full solution

2
(b) Reflection over $y$ -axis: negate the input → $h(x)=f(-x)=(-x)^3-2=-x^3-2$ . $h(2)=-8-2=-10$ .
3
Note: $f(2)=8-2=6$ ; the $x$ -axis reflection negates the output ( $-6$ ); the $y$ -axis reflection changes the sign of $x$ ( $-10$ ).

Two fundamental reflections:

-f(x)

flips the graph over the

x

-axis (negates all outputs);

f(-x)

flips over the

y

-axis (reverses all inputs). They are distinct transformations that generally produce different results.

Example 2

medium

Show that

f(x)=x^2

is unchanged by reflection over the

y

-axis (even function) but

f(x)=x^3

is negated by this reflection (odd function).

Example 3

medium

For

f(x) = \sqrt{x}

, write

f(-x)

and give its domain.

Example 4

medium

Classify

f(x) = \sin(x)

as even, odd, or neither.

Example 5

medium

f(x) = x^2 - 6x + 8

, find the vertex of

-f(x)

Example 6

hard

Show that the only function that is both even and odd is

f(x) = 0

Example 7

hard

Decompose

f(x) = e^x

into even and odd parts using

f_e = \frac{f(x)+f(-x)}{2}

f_o = \frac{f(x)-f(-x)}{2}

Example 8

hard

f

is odd and integrable on

[-a, a]

. Show

\int_{-a}^{a} f(x)\,dx = 0

Example 9

challenge

Find all

x

where

f(x) = x^3 - 4x

meets its

y

-axis reflection

g(x) = f(-x)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

The point

(-3, 7)

is on the graph of

y=f(x)

. Give the corresponding point on: (a)

y=-f(x)

, (b)

y=f(-x)

, (c)

y=-f(-x)

Example 2

hard

Classify

f(x) = x^3 + x

and

g(x) = x^4 + x^2 + 1

as even, odd, or neither. Explain using the definitions.

Example 3

easy

Describe the transformation from

f(x)

-f(x)

Example 4

easy

Describe the transformation from

f(x)

f(-x)

Example 5

easy

f(x)

has the point

(3, 4)

, where does it go under

-f(x)

Example 6

easy

f(x)

has the point

(3, 4)

, where does it go under

f(-x)

Example 7

easy

For

f(x) = x^2 + 1

, write

-f(x)

explicitly.

Example 8

easy

For

f(x) = 2x + 3

, write

f(-x)

explicitly.

Example 9

easy

The x-intercept of

f

is at

x = 5

. Where is the x-intercept of

-f(x)

Example 10

easy

A function satisfies

f(-x) = f(x)

for all

x

. What symmetry does its graph have?

Example 11

medium

f(x) = x^3

even, odd, or neither? Justify using

f(-x)

Example 12

medium

f(x) = x^2 + x

even, odd, or neither? Justify.

Example 13

medium

The graph

y = \sqrt{x}

starts at the origin going right. Write the equation of its reflection over the y-axis and state its domain.

Example 14

medium

Write the function that reflects

f(x) = x^2 - 4x

over the x-axis, then give its vertex.

Example 15

medium

g(x) = f(-x)

and

f

passes through

(-2, 7)

and

(4, -1)

, list the points

g

passes through.

Example 16

medium

The inverse of a one-to-one function reflects its graph over the line

y = x

. If

f

passes through

(2, 9)

, what point must

f^{-1}

pass through?

Example 17

medium

A graph is symmetric about the origin. If it contains

(3, 5)

, what other point must it contain, and what does this say about

f(-x)

Example 18

challenge

Show that reflecting

f(x)

over the x-axis and then over the y-axis gives the same result as reflecting over the origin, and write the final expression in terms of

f

Example 19

challenge

Find all values where

f(x) = x^2 - 9

and its x-axis reflection

-f(x)

intersect.

Example 20

challenge

Decompose

f(x) = x^2 + x^3

into an even part and an odd part using

f_{even}(x) = \frac{f(x)+f(-x)}{2}

and

f_{odd}(x) = \frac{f(x)-f(-x)}{2}

Example 21

medium

Determine whether

f(x) = x^4 - 3x^2

is even, odd, or neither, and name its symmetry.

Example 22

medium

g(x) = -f(x)

and

f

has a minimum of

-2

x = 3

, what feature does

g

have at

x = 3

Example 23

easy

For

f(x) = 3x - 2

, write

-f(x)

Example 24

easy

Point

(2, -7)

is on

y = f(x)

. Find its image on

y = -f(x)

Example 25

easy

For

f(x) = x^2

, is

f(x) = f(-x)

? If so, name the symmetry.

Example 26

easy

f

has

y

-intercept

(0, 5)

. What is the

y

-intercept of

-f(x)

Example 27

medium

Classify

f(x) = x^5 - x

as even, odd, or neither.

Example 28

medium

Classify

f(x) = \cos(x)

as even, odd, or neither.

Example 29

medium

f

has roots at

x = 1, 4

. What are the roots of

f(-x)

Example 30

medium

f

passes through

(1, 3), (2, 5), (3, 7)

. List the points on

-f(-x)

Example 31

hard

f

is even and

g

is odd, is

f \cdot g

even, odd, or neither?

Example 32

hard

f(x) = \ln(x)

. Write

f(-x)

and state its domain.

Example 33

hard

f(x) = (x-2)^2

. Write the function obtained by reflecting over the

y

-axis.

Example 34

hard

f

is even and

f(3) = 7

, find

f(-3)

and

-f(3)

Example 35

challenge

f(x) = |x|

. Is

f

even, odd, or neither? Justify.

Example 36

challenge

f(x) = \tan(x)

. Classify as even, odd, or neither.

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Related Concepts

Function Transformation Even and Odd Functions Symmetry

Background Knowledge

These ideas may be useful before you work through the harder examples.

transformation