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 xx-axis (−f(x)-f(x)), yy-axis (f(−x)f(-x)), or the line y=xy = x (the inverse function).

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

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

challenge
Show that reflecting f(x)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 ff.

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

medium
f(x)=exf(x) = e^x. Write f(−x)f(-x) in simplest form.

Example 10

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

Example 11

hard
Classify f(x)=x3+xf(x) = x^3 + x and g(x)=x4+x2+1g(x) = x^4 + x^2 + 1 as even, odd, or neither. Explain using the definitions.

Example 12

hard
ff is odd and integrable on [−a,a][-a, a]. Show ∫−aaf(x) dx=0\int_{-a}^{a} f(x)\,dx = 0.

Example 13

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

Example 14

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

Example 15

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

Example 16

easy
Given f(x)=x3−2f(x)=x^3-2, write the equations for (a) reflection over the xx-axis and (b) reflection over the yy-axis. Evaluate each at x=2x=2.

Example 17

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

Example 18

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

Example 19

challenge
Find all values where f(x)=x2−9f(x) = x^2 - 9 and its x-axis reflection −f(x)-f(x) intersect.

Example 20

easy
The point (−3,7)(-3, 7) is on the graph of y=f(x)y=f(x). Give the corresponding point on: (a) y=−f(x)y=-f(x), (b) y=f(−x)y=f(-x), (c) y=−f(−x)y=-f(-x).
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