Chapter T, Problem 7CDT

If f(x) = x² + 2x − 1 and g(x) = 2x − 3, find each of the following functions.

1. (a) f ∘ g
2. (b) g ∘ f
3. (c) g ∘ g ∘ g

To determine

To find: The value of $f\circ g$.

Answer to Problem 7CDT

The value of $f\circ g$ is $4x^2-8x+2$.

Explanation of Solution

Given;

The functions are $f\left(x\right)=x^2+2x-1$ and $g\left(x\right)=2x-3$.

Calculation:

Consider the given functions $f\left(x\right)=x^2+2x-1$ and $g\left(x\right)=2x-3$.

The composition of the functions is,

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

Substitute $g\left(x\right)=2x-3$ in $f\left(g\left(x\right)\right)$ to find the value of $f\circ g$,

$f\left(g\left(x\right)\right)=f\left(2x-3\right)$

Substitute $x=2x-3$ in $f\left(x\right)=x^2+2x-1$ to find the value of $f\left(2x-3\right)$,

$\begin{array}{c}f\left(2x-3\right)={\left(2x-3\right)}^2+2\times \left(2x-3\right)-1\\ =4x^2-12x+9+4x-6-1\\ =4x^2-8x+2\end{array}$

Thus, the value of $f\circ g$ is $4x^2-8x+2$.

To determine

To find: The value of $g\circ f$.

Answer to Problem 7CDT

The value of $g\circ f$ is $2x^2+4x-5$.

Explanation of Solution

Given;

The functions are $f\left(x\right)=x^2+2x-1$ and $g\left(x\right)=2x-3$.

Calculation:

Consider the given functions $f\left(x\right)=x^2+2x-1$ and $g\left(x\right)=2x-3$.

The composition of the functions is,

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

Substitute $g\left(x\right)=2x-3$ in $g\left(f\left(x\right)\right)$ to find the value of $g\circ f$,

$g\left(f\left(x\right)\right)=g\left(x^2+2x-1\right)$

Substitute $x=x^2+2x-1$ in $g\left(x\right)=2x-3$ to find the value of $g\left(x^2+2x-1\right)$,

$\begin{array}{c}g\left(x^2+2x-1\right)=2\left(x^2+2x-1\right)-3\\ =2x^2+4x-2-3\\ =2x^2+4x-5\end{array}$

Thus, the value of $g\circ f$ is $2x^2+4x-5$.

To determine

To find: The value of $g\circ g\circ g$.

Answer to Problem 7CDT

The value of $g\circ g\circ g$ is $8x-21$.

Explanation of Solution

Given;

The functions are $f\left(x\right)=x^2+2x-1$ and $g\left(x\right)=2x-3$.

Calculation:

Consider the given functions $f\left(x\right)=x^2+2x-1$ and $g\left(x\right)=2x-3$.

The composition of the functions is,

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

Substitute $g\left(x\right)=2x-3$ in $f\left(g\left(x\right)\right)$ to find the value of $g\circ g\circ g$,

$g\left(g\left(g\left(x\right)\right)\right)=g\left(g\left(2x-3\right)\right)$

Substitute $x=2x-3$ in $g\left(x\right)=2x-3$ to find the value of $g\left(2x-3\right)$,

$\begin{array}{c}g\left(2x-3\right)=2\left(2x-3\right)-3\\ =4x-6-3\\ =4x-9\end{array}$

Substitute $x=2x-3$ in $g\left(x\right)=2x-3$ to find the value of $g\left(4x-9\right)$,

$\begin{array}{c}g\left(4x-9\right)=2\left(4x-9\right)-3\\ =8x-18-3\\ =8x-21\end{array}$

Thus, the value of $g\circ g\circ g$ is $8x-21$.