Reflecting Functions

Functions
process

Also known as: function reflection, flip over axis, mirror image

Grade 9-12

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Reflecting a function mirrors its graph across the x-axis (-f(x)), y-axis (f(-x)), or the line y = x (the inverse function). Reflections reveal the relationship between a function and its inverse (reflection across y = x) and connect even/odd symmetry to the function's formula.

This concept is covered in depth in our Functions and Graphs Guide, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

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

Example

f(x) = x^2 \to -f(x) = -x^2 (opens down).
f(-x) = (-x)^2 = x^2 (unchanged—symmetric).

Formula

-f(x) reflects over x-axis; f(-x) reflects over y-axis

Notation

Even function: f(-x) = f(x) (symmetric about y-axis). Odd function: f(-x) = -f(x) (symmetric about origin).

🌟 Why It Matters

Reflections reveal the relationship between a function and its inverse (reflection across y = x) and connect even/odd symmetry to the function's formula.

💭 Hint When Stuck

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.

Formal View

-f(x): (x, y) \mapsto (x, -y) (reflect over x-axis). f(-x): (x, y) \mapsto (-x, y) (reflect over y-axis). Even: f(-x) = f(x)\;\forall x. Odd: f(-x) = -f(x)\;\forall x

🚧 Common Stuck Point

Even functions: f(-x) = f(x). Odd functions: f(-x) = -f(x).

⚠️ Common Mistakes

  • Confusing -f(x) with f(-x) — -f(x) reflects over the x-axis (flips output sign); f(-x) reflects over the y-axis (flips input sign)
  • Thinking reflection changes the shape of the graph — reflection only flips the graph; it preserves the shape exactly
  • Forgetting to check for symmetry — if f(-x) = f(x) the function is even (symmetric about y-axis); if f(-x) = -f(x) it is odd (symmetric about origin)

Frequently Asked Questions

What is Reflecting Functions in Math?

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

Why is Reflecting Functions important?

Reflections reveal the relationship between a function and its inverse (reflection across y = x) and connect even/odd symmetry to the function's formula.

What do students usually get wrong about Reflecting Functions?

Even functions: f(-x) = f(x). Odd functions: f(-x) = -f(x).

What should I learn before Reflecting Functions?

Before studying Reflecting Functions, you should understand: transformation.

Prerequisites

transformation

How Reflecting Functions Connects to Other Ideas

To understand reflecting functions, you should first be comfortable with transformation. Once you have a solid grasp of reflecting functions, you can move on to even odd functions and symmetry.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Functions and Graphs: Complete Foundations for Algebra and Calculus →
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