Reflecting Functions Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Classify

f(x) = x^3 + x

and

g(x) = x^4 + x^2 + 1

as even, odd, or neither. Explain using the definitions.

1
$f(-x) = (-x)^3+(-x) = -x^3-x = -(x^3+x)=-f(x)$ . So $f$ is odd.
2
$g(-x) = (-x)^4+(-x)^2+1 = x^4+x^2+1 = g(x)$ . So $g$ is even.
3
A function with all odd powers (and zero constant) is odd; a function with all even powers (and constants) is even. $f$ has only odd powers; $g$ has only even powers and a constant.

Answer

f(x)=x^3+x

is odd;

g(x)=x^4+x^2+1

is even

To classify, substitute

-x

and simplify. Polynomials built from only odd-power terms are odd functions; those built from only even-power terms (including constants, which are

x^0

) are even functions.

About Reflecting Functions

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

Given [formula], write the equations for (a) reflection over the [formula]-axis and (b) reflection o

Show that [formula] is unchanged by reflection over the [formula]-axis (even function) but [formula]

The point [formula] is on the graph of [formula]. Give the corresponding point on: (a) [formula], (b