Chapter 1.3, Problem 42E

To determine

To find: $\left(f\circ g\right)\left(-3\right)$ and $\left(g\circ f\right)\left(-3\right)$ , if $f\left(x\right)=x^2-4x+5$ and $g\left(x\right)=x-2$

Answer to Problem 42E

  $\begin{array}{c}\left(f\circ g\right)\left(-3\right)=50\\ \left(g\circ f\right)\left(-3\right)=24\end{array}$

Explanation of Solution

Given information: The given functions are as follows:

  $f\left(x\right)=x^2-4x+5$

  $g\left(x\right)=x-2$

Calculation:

The formula for $\left(f\circ g\right)\left(x\right)$ is as follows:

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ ={\left(g\left(x\right)\right)}^2-4\left(g\left(x\right)\right)+5\left[\because f\left(x\right)=x^2-4x+5\right]\\ ={\left(x-2\right)}^2-4\left(x-2\right)+5\left[\text{Substituting }g\left(x\right)=x-2\right]\\ =x^2-4x+4-4x+8+5\\ =x^2-8x+17\end{array}$

Now substitute $x=-3$ , in the above equation and calculate $\left(f\circ g\right)\left(-3\right)$ as shown below.

  $\begin{array}{c}\left(f\circ g\right)\left(-3\right)=x^2-8x+17\\ ={\left(-3\right)}^2-8\left(-3\right)+17\\ =9+24+17\\ =50\end{array}$

Thus, $\left(f\circ g\right)\left(-3\right)$ is 50.

The formula for $\left(f\circ g\right)\left(x\right)$ is as follows:

  $\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\\ =\left(f\left(x\right)\right)-2\left[\because g\left(x\right)=x-2\right]\\ =\left(x^2-4x+5\right)-2\left[\text{Substituting }f\left(x\right)=x^2-4x+5\right]\\ =x^2-4x+3\end{array}$

Now substitute $x=-3$ , in the above equation and calculate $\left(g\circ f\right)\left(-3\right)$ as shown below.

  $\begin{array}{c}\left(g\circ f\right)\left(-3\right)=x^2-4x+3\\ ={\left(-3\right)}^2-4\left(-3\right)+3\\ =9+12+3\\ =24\end{array}$

Thus, $\left(g\circ f\right)\left(-3\right)$ is 24.