Question

if the function $f(x) = mx + b$ has an inverse function, which statement must be true?
$\bigcirc$ $m \
eq 0$
$\bigcirc$ $m = 0$
$\bigcirc$ $b \
eq 0$
$\bigcirc$ $b = 0$

Explanation:

Step1: Recall inverse function condition

A function has an inverse if and only if it is one-to-one (strictly monotonic). For a linear function $f(x)=mx+b$, this requires it to be non-constant and strictly increasing/decreasing.

Step2: Analyze linear function slope

A linear function is constant if its slope $m=0$ (it becomes $f(x)=b$, a horizontal line, which fails the horizontal line test for invertibility). Thus, the slope cannot be 0.

Step3: Evaluate intercept $b$

The value of $b$ (the y-intercept) does not affect whether the function is one-to-one. A linear function with any $b$ (as long as $m
eq0$) will still be strictly monotonic and have an inverse.

Answer:

A. $m
eq 0$