Even Functions

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Idea

An even function looks the same after reflecting its graph across the vertical axis. Its domain must be symmetric about zero: whenever $x$ is allowed, $-x$ must be allowed too.

Method

f(-x)=f(x).

Test the whole formula, not merely its visible exponents. Absolute value and cosine are even; many expressions containing odd powers are neither even nor odd.

Worked example

For $f(x)=x^4-3x^2+2$ , substitution gives $f(-x)=(-x)^4-3(-x)^2+2=f(x)$ . For $g(x)=x^2+x$ , $g(-x)=x^2-x$ is generally different. Restricting $x^2$ to $[0,\infty)$ removes the symmetric domain, so the restricted function is not even under the standard definition.

Common mistake

Matching $f(1)$ and $f(-1)$ alone does not prove evenness; the equality must hold throughout the domain.

Check your understanding

Is $f(x)=1/(1+x^2)$ even?

Show answer

Yes, its domain is all real numbers and substituting $-x$ leaves the formula unchanged.

Explore

Domain and range

Try this. Switch functions and compare allowed inputs and outputs. Open circles exclude endpoints; filled circles include them.

Function

y = x² · Domain: all real x · Range: y ≥ 0

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which function is even on ℝ?

f(x)=x+1

f(x)=x²+1

f(x)=x³

Hint 1 · Find a starting point

An even function satisfies f(−x)=f(x).

Hint 2 · Take the next step

Substitute −x into each expression.

Show the reasoning

Answer: f(x)=x²+1

(−x)²+1=x²+1, giving symmetry about the y-axis.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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