Chapter 2.5, Problem 60E

To determine

To find: $\left[f\cdot g\right]\left(x\right)$ and $\left[g\cdot f\right]\left(x\right)$

Answer to Problem 60E

  $\begin{array}{l}\left[f\cdot g\right]\left(x\right)=x\left(x+1\right)\\ \left[g\cdot f\right]\left(x\right)=x^2+3x+1\end{array}$

Explanation of Solution

Given information:

The given functions are as follows:

  $\begin{array}{l}f\left(x\right)=x^2+3x+2\\ g\left(x\right)=x-1\end{array}$

Calculation:

Solve the function as follows:

  $\begin{array}{c}\left[f\cdot g\right]\left(x\right)=f\left(x-1\right)\\ ={\left(x-1\right)}^2+3\left(x-1\right)+2\\ =x^2-2x+1+3x-3+2\\ =x^2+x\\ =x\left(x+1\right)\left[\text{Taking Common factor}\right]\end{array}$

  $\begin{array}{c}\left[g\cdot f\right]\left(x\right)=g\left(x^2+3x+2\right)\\ =\left(x^2+3x+2\right)-1\\ =x^2+3x+2-1\\ =x^2+3x+1\end{array}$

Thus, the value of $\left[f\cdot g\right]\left(x\right)$ and $\left[g\cdot f\right]\left(x\right)$ is $x\left(x+1\right)$ and $x^2+3x+1$ respectively.