Practice Reflecting Functions in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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).

Showing a random 20 of 50 problems.

Example 1

medium
Classify f(x)=sin⁡(x) as even, odd, or neither.

Example 2

easy
f has y-intercept (0,5). What is the y-intercept of −f(x)?

Example 3

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 4

easy
If f(x) has the point (3,4), where does it go under f(−x)?

Example 5

challenge
f(x)=tan⁡(x). Classify as even, odd, or neither.

Example 6

easy
Describe the transformation from f(x) to f(−x).

Example 7

medium
f has roots at x=1,4. What are the roots of f(−x)?

Example 8

medium
If g(x)=−f(x) and f has a minimum of −2 at x=3, what feature does g have at x=3?

Example 9

medium
f(x)=ex. Write f(−x) in simplest form.

Example 10

easy
For f(x)=3x−2, write f(−x).

Example 11

hard
Classify f(x)=x3+x and g(x)=x4+x2+1 as even, odd, or neither. Explain using the definitions.

Example 12

hard
f is odd and integrable on [−a,a]. Show ∫−aaf(x) dx=0.

Example 13

easy
For f(x)=2x+3, write f(−x) explicitly.

Example 14

medium
If g(x)=f(−x) and f passes through (−2,7) and (4,−1), list the points g passes through.

Example 15

medium
Classify f(x)=cos⁡(x) as even, odd, or neither.

Example 16

easy
Given f(x)=x3−2, write the equations for (a) reflection over the x-axis and (b) reflection over the y-axis. Evaluate each at x=2.

Example 17

hard
If f is even and f(3)=7, find f(−3) and −f(3).

Example 18

easy
A function satisfies f(−x)=f(x) for all x. What symmetry does its graph have?

Example 19

challenge
Find all values where f(x)=x2−9 and its x-axis reflection −f(x) intersect.

Example 20

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).
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