Chapter 11, Problem 1PCT

Given the functions f(x) = 2x + 3 and g(x) = 2x² – 4x, find

1. a. (f ∘ g)(x)
2. b. (g ∘ f)(x)
3. c. (f ∘ g)(3)
4. d. (g ∘ f)(−2)
5. e. (f ∘ f)(1)

To determine

The composite function $\left(f\circ g\right)\left(x\right)$ for the functions $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

Answer to Problem 1PCT

The value of the composite function $\left(f\circ g\right)\left(x\right)$ is $4x^2-8x+3$.

Explanation of Solution

Definition used:

Given two functions f and g, the composite function, denoted by $f\circ g$ (read as f composed with g), is defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$. The notation $f\left(g\left(x\right)\right)$ is read f of g of x.

Calculation:

The given functions are, $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

By using the above definition, find the value of the composite function $\left(f\circ g\right)\left(x\right)$ as follows.

$\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\left(\because \left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\right)\\ =f\left(2x^2-4x\right)\left(\because g\left(x\right)=2x^2-4x\right)\\ =2\left(2x^2-4x\right)+3\left(\because f\left(x\right)=2x+3\right)\\ =4x^2-8x+3\end{array}$

Thus, the value of the composite function $\left(f\circ g\right)\left(x\right)$ is $4x^2-8x+3$.

To determine

The composite function $\left(g\circ f\right)\left(x\right)$ for the functions $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

Answer to Problem 1PCT

The value of the composite function $\left(g\circ f\right)\left(x\right)$ is $8x^2+16x+6$.

Explanation of Solution

The given functions are $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

By using the above definition, find the value of the composite function $\left(g\circ f\right)\left(x\right)$ as follows.

$\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\left(\because \left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\right)\\ =g\left(2x+3\right)\left(\because f\left(x\right)=2x+3\right)\\ =2{\left(2x+3\right)}^2-4\left(2x+3\right)\left(\because g\left(x\right)=2x^2-4x\right)\\ =8x^2+24x+18-8x-12\end{array}$

$=8x^2\text{+16}x+6\left(\text{Combine like terms}\right)$

Thus, the value of the composite function $\left(g\circ f\right)\left(x\right)$ is $8x^2+16x+6$.

To determine

The composite function $\left(f\circ g\right)\left(3\right)$ for the functions $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

Answer to Problem 1PCT

The value of $\left(f\circ g\right)\left(3\right)$ is 15.

Explanation of Solution

From part (a) it is observed that, the value of the composite function $\left(f\circ g\right)\left(x\right)$ is $4x^2-8x+3$.

Substitute $x=3$ in $\left(f\circ g\right)\left(x\right)=4x^2-8x+3$.

$\begin{array}{c}\left(f\circ g\right)\left(3\right)=4{\left(3\right)}^2-8\left(3\right)+3\\ =36-24+3\\ =15\end{array}$

Thus, the value of $\left(f\circ g\right)\left(3\right)$ is 15.

To determine

The composite function $\left(g\circ f\right)\left(-2\right)$ for the functions $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

Answer to Problem 1PCT

The value of $\left(g\circ f\right)\left(-2\right)$ is 6.

Explanation of Solution

From part (b) it is observed that, the value of the composite function $\left(g\circ f\right)\left(x\right)$ is $8x^2+16x+6$.

Substitute $x=-2$ in $\left(g\circ f\right)\left(x\right)=8x^2+16x+6$.

$\begin{array}{c}\left(g\circ f\right)\left(-2\right)=8{\left(-2\right)}^2\text{+16}\left(-2\right)\text{+6}\\ =\text{32}-\text{32}+\text{6}\\ =\text{6}\end{array}$

Thus, the value of $\left(g\circ f\right)\left(-2\right)$ is 6.

To determine

The composite function $\left(f\circ f\right)\left(1\right)$ for the functions $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

Answer to Problem 1PCT

The value of the composite function $\left(f\circ f\right)\left(1\right)$ is $13$.

Explanation of Solution

The given functions are, $f\left(x\right)=2x+3$ and $g\left(x\right)=2x^2-4x$.

By using the above definition, find the value of the composite function $\left(f\circ f\right)\left(x\right)$ as follows.

$\begin{array}{c}\left(f\circ f\right)\left(x\right)=f\left(f\left(x\right)\right)\left(\because \left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\right)\\ =f\left(2x+3\right)\left(\because f\left(x\right)=2x+3\right)\\ =2\left(2x+3\right)\text{+3 }\left(\because f\left(x\right)=2x+7\right)\\ =4x+6+3\end{array}$

$=4x+9\left(\text{Combine like terms}\right)$

That is, $\left(f\circ f\right)\left(x\right)=4x+9$.

Substitute $x=1$ in $\left(f\circ f\right)\left(x\right)$ to obtain the value of $\left(f\circ f\right)\left(1\right)$ as follows.

$\begin{array}{c}\left(f\circ f\right)\left(x\right)=4x+9\\ \left(f\circ f\right)\left(1\right)=4\left(1\right)+9\\ =4+9\\ =13\end{array}$

Thus, the value of the composite function $\left(f\circ f\right)\left(1\right)$ is $13$.