Chapter 1.5, Problem 43E

(a)

To determine

To calculate: To determine $f\circ g$

Answer to Problem 43E

  $f\circ g=20-3x$

Explanation of Solution

Given:

  $f(x)=3x+5$

  $g(x)=5-x$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows:

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Calculation:

  $f\circ g$ is calculated as:

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

Substitute: $g(x)=5-x$

  $\left(f\circ g\right)\left(x\right)$ = $f(5-x)$

Also, given $f(x)=3x+5$

Thus, $\left(f\circ g\right)\left(x\right)$ is calculated as

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=3(5-x)+5\\ \left(f\circ g\right)\left(x\right)=15-3x+5\\ \left(f\circ g\right)\left(x\right)=20-3x\end{array}$

Conclusion:

Hence, $\left(f\circ g\right)\left(x\right)=20-3x$

(b)

To determine

To calculate: To determine $g\circ f$

Answer to Problem 43E

  $g\circ f=-3x$

Explanation of Solution

Given:

  $f(x)=3x+5$

  $g(x)=5-x$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows:

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Calculation:

Equation of $g\circ f$ is calculated as:

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

Substitute: $f(x)=3x+5$

Thus,

  $\left(g\circ f\right)\left(x\right)=g\left(3x+5\right)$

Also given that $g(x)=5-x$

Thus, $g\circ f$ is calculated as follows:

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

Conclusion:

Hence, $\left(g\circ f\right)\left(x\right)=-3x$

(c)

To determine

To calculate: To determine $\left(f\circ g\right)\left(0\right)$

Answer to Problem 43E

  $\left(f\circ g\right)\left(0\right)=20$

Explanation of Solution

Given:

  $f(x)=3x+5$

  $g(x)=5-x$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows: $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Calculation:

From part (a):

  $\left(f\circ g\right)\left(x\right)=20-3x$

To calculate $\left(f\circ g\right)\left(0\right)$ replace $x$ with 0:

  $\begin{array}{l}\left(f\circ g\right)\left(0\right)=20-3(0)\\ \left(f\circ g\right)\left(0\right)=20-0\\ \left(f\circ g\right)\left(0\right)=20\end{array}$

Conclusion:

Hence, $\left(f\circ g\right)\left(0\right)=20$