Chapter 7.7, Problem 85STP

To determine

To Find :

The inverse function.

Answer to Problem 85STP

  $y=\frac{-x-3}{2}$

Explanation of Solution

Given Information:

Given function is:

  $y=-2x-3$

Concept Used:

Inverse Function :

Two functions f and g are inverse functions if and only if both of their compositions are the identity function.

That is ,

f(x) and g(x) are inverses if and only if $[f\circ g](x)=x\text{ and }[g\circ f](x)=x$ .

Calculation:

  $\begin{array}{l}Given\\ y=-2x-3=f(x)\\ \text{Interchange x and y , then solve for y to get }\\ \text{the inverse function:}\\ x=-2y-3\\ \Rightarrow x+3=-2y\mathrm{............}[\text{add 3 each side]}\\ \Rightarrow y=\frac{x+3}{-2}\mathrm{............}[\text{divide each side by -2]}\\ \Rightarrow y=\frac{-x-3}{2}\\ \Rightarrow g(x)=\frac{-x-3}{2}\\ Hence,\text{ f(x) and g(x) are inverse functions}\text{.}\\ \text{Check:}\\ \left[f\circ g\right](x)=f(g(x))=f\left(\frac{-x-3}{2}\right)=-2\left(\frac{-x-3}{2}\right)-3=(x+3)-3=x\\ \left[g\circ f\right](x)=g(f(x))=g\left(-2x-3\right)=\frac{-(-2x-3)-3}{2}=\frac{2x}{2}=x\\ So,\text{ f and g are inverse functions}\text{.}\end{array}$