Q. 55 - Precalculus Enhanced with Graphing Utilities

Question

Under what circumstances is a linear function $f\left(x\right)=mx+b$ odd? Can a linear function ever be even?

Step-by-Step Solution

Answer

A linear function $f\left(x\right)=mx+b$ will be odd if and only if $b=0$

A linear function $f\left(x\right)=mx+b$ will be an even function if the slope $m=0$and it is a constant function.

Step 1. Given information

The given linear function is

$f\left(x\right)=mx+b$

Step 2. Odd function

If the y-intercept of a linear function $f\left(x\right)=mx+b$ then the line will pass through the origin

$$\begin{gathered} f\left(x\right)=mx+b \\ f\left(x\right)=mx \\ f(-x)=m\left(-x\right) \\ f(-x)=-mx \\ f(-x)=-f\left(x\right) \end{gathered}$$

So a linear function will be an odd function if its y-intercept  $b=0$

Step 3. Even function

A liner will be a constant function if its slope $m=0$

$$\begin{gathered} f\left(x\right)=mx+b \\ f\left(x\right)=b \\ f(-x)=b \\ f(-x)=f\left(x\right) \end{gathered}$$

So a linear function will even function if its slope $m=0$