Reflecting Functions Math Example 2

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

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).

1
For $f(x)=x^2$ : $f(-x)=(-x)^2=x^2=f(x)$ . Reflection over $y$ -axis maps the graph to itself. This is the definition of an even function.
2
For $f(x)=x^3$ : $f(-x)=(-x)^3=-x^3=-f(x)$ . Reflection over $y$ -axis is equivalent to also reflecting over the $x$ -axis. This is the definition of an odd function.
3
Geometrically: even functions are symmetric about the $y$ -axis; odd functions have $180°$ rotational symmetry about the origin.

Answer

x^2

is even (

f(-x)=f(x)

);

x^3

is odd (

f(-x)=-f(x)

)

Even and odd functions are defined by their symmetry under reflection. Even functions are symmetric about the

y

-axis; odd functions are symmetric about the origin (equivalent to reflecting over both axes).

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

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

Classify [formula] and [formula] as even, odd, or neither. Explain using the definitions.