Chapter 1.2, Problem 6CFU

To determine

To find: $\left[f\circ g\right]\left[x\right]$ and $\left[g\circ f\right]\left[x\right]$.

Answer to Problem 6CFU

  $\begin{array}{l}\left[f\circ g\right]\left[x\right]=2x+11\\ \left[g\circ f\right]\left[x\right]=2x+8\end{array}$

Explanation of Solution

Given information: The functions given are as follows:

  $\begin{array}{l}f\left(x\right)=2x+5\\ g\left(x\right)=3+x\end{array}$

Calculation:

  $\left[f\circ g\right]\left[x\right]=f\left(g\left(x\right)\right)$

Substitute $g\left(x\right)$ into the function $f\left(x\right)$ as shown below.

  $\begin{array}{l}f\left(x\right)=2x+5\\ f\left(g\left(x\right)\right)=2\left(g\left(x\right)\right)+5\end{array}$

Now substitute the value of $g\left(x\right)$ in the above equation and simplify as follows:

  $\begin{array}{l}f\left(3+x\right)=2\left(3+x\right)+5\\ f\left(3+x\right)=6+2x+5\\ f\left(3+x\right)=2x+11\\ \left[f\circ g\right]\left[x\right]=2x+11\text{ [Rewrite the function in}\left(f\circ g\right)\left(x\right)\text{ ]}\end{array}$

Therefore, $\left[f\circ g\right]\left[x\right]$ is $2x+11$ .

Similarly, $\left[g\circ f\right]\left[x\right]=g\left(f\left(x\right)\right)$

Substitute $g\left(x\right)$ into the function $f\left(x\right)$ as shown below.

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

Now substitute the value of $g\left(x\right)$ in the above equation and simplify as follows:

  $\begin{array}{l}g\left(2x+5\right)=3+\left(2x+5\right)\\ g\left(2x+5\right)=2x+8\\ \left[g\circ f\right]\left[x\right]=2x+8\text{ [Rewrite the function in}\left(g\circ f\right)\left(x\right)\text{ ]}\end{array}$

Therefore, $\left[g\circ f\right]\left[x\right]$ is $2x+8$ .