Question

which function has an inverse that is a function?
$b(x) = x^2 + 3$
$d(x) = -9$
$m(x) = -7x$
$p(x) = |x|$

Explanation:

Step1: Recall inverse function rule

A function has an inverse that is a function if and only if it is one-to-one (passes the Horizontal Line Test: no horizontal line intersects the graph more than once).

Step2: Analyze $b(x)=x^2+3$

This is a parabola opening upwards. For example, $b(1)=4$ and $b(-1)=4$, so it fails the Horizontal Line Test.

Step3: Analyze $d(x)=-9$

This is a horizontal line. Every input maps to -9, so it fails the Horizontal Line Test.

Step4: Analyze $m(x)=-7x$

This is a linear function with a non-zero slope ($-7$). Linear functions with non-zero slopes are one-to-one, so it passes the Horizontal Line Test.

Step5: Analyze $p(x)=|x|$

This V-shaped graph has $p(1)=1$ and $p(-1)=1$, so it fails the Horizontal Line Test.

Answer:

$m(x) = -7x$