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

Explanation:

Step1: Recall the condition for inverse to be a function

A function has an inverse that is also a function if and only if the original function is one - to - one. A one - to - one function passes the horizontal line test, i.e., any horizontal line intersects the graph of the function at most once.

Step2: Analyze (g(x)=2x - 3)

(g(x)=2x - 3) is a linear function. The slope of (y = 2x-3) is (m = 2\neq0). For a non - vertical and non - horizontal linear function, any horizontal line will intersect its graph at exactly one point. So it is one - to - one.

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

(k(x)=-9x^{2}) is a quadratic function. Its graph is a parabola opening downwards ((y=-9x^{2}), (a=-9<0)). A horizontal line (y = c) ((c\leq0)) will intersect the parabola at two points. So it is not one - to - one.

Step4: Analyze (f(x)=\vert x + 2\vert)

The graph of (y=\vert x + 2\vert) is a V - shaped graph with the vertex at ((-2,0)). A horizontal line (y = c>0) will intersect the graph at two points. So it is not one - to - one.

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

(w(x)=-20) is a constant function. A horizontal line (y=-20) will intersect the graph of the function at infinitely many points. So it is not one - to - one.

Answer:

A. (g(x)=2x - 3)