which function has an inverse that is also a function?\n$g(x)=2x-3$\n$k(x)=-9x^2$\n$f(x)=|x + 2|$\n$w(x)=-20$

Explanation:

Step1: Recall inverse function rule

A function has an inverse that is also 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 $g(x)=2x-3$

This is a linear function with a non-zero slope ($m=2$). Linear functions with non-zero slopes are strictly increasing/decreasing, so they pass the Horizontal Line Test.

Step3: Analyze $k(x)=-9x^2$

This is a downward-opening parabola. A horizontal line will intersect it at two points (e.g., $y=-9$ intersects at $x=1$ and $x=-1$), so it fails the Horizontal Line Test.

Step4: Analyze $f(x)=|x+2|$

This is a V-shaped absolute value function. A horizontal line above the vertex will intersect it at two points (e.g., $y=1$ intersects at $x=-1$ and $x=-3$), so it fails the Horizontal Line Test.

Step5: Analyze $w(x)=-20$

This is a horizontal constant function. A horizontal line (itself) intersects it at infinitely many points, so it fails the Horizontal Line Test.

Answer:

$g(x) = 2x - 3$