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: Negative outside = flip over x-axis. Negative inside = flip over y-axis.

Common stuck point: Even functions: f(-x) = f(x). Odd functions: f(-x) = -f(x).

Sense of Study hint: Compute f(-x) and compare it to f(x). If they are equal, the function is even. If f(-x) = -f(x), it is odd. If neither, it is neither.

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.

Solution

  1. 1
    (a) Reflection over x-axis: negate the output → g(x)=-f(x)=-(x^3-2)=-x^3+2. g(2)=-8+2=-6.
  2. 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. 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).

Answer

(a) g(x)=-x^3+2, g(2)=-6; (b) h(x)=-x^3-2, h(2)=-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).

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.
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Background Knowledge

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

transformation