Question

Evaluating Composition of Functions Use $f\left(x\right)=2x-3$and $g\left(x\right)=4-x^2$to evaluate the expression.

1. $\left(f\circ g\right)\left(-2\right)$
   
2. $\left(g\circ f\right)\left(-2\right)$
   

Short answer

1. The value of$\left(f\circ g\right)\left(-2\right)$ is -3.
2. The value of$\left(g\circ f\right)\left(-2\right)$ is -45.

Step-by-step solution

(a)Step 1 – Composition of functions.

Composition of functions is defined as:

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

– Given information.

The given functions are:

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

– Find the required value.

$\begin{array}{c}\left(f\circ g\right)\left(-2\right)=f\left(g\left(-2\right)\right)\\ =f\left(4-{\left(-2\right)}^2\right)\left[\because g\left(x\right)=4-x^2\right]\\ =f\left(4-4\right)\\ =f\left(0\right)\\ =2\left(0\right)-3\left[\because f\left(x\right)=2x-3\right]\\ =0-3\\ =-3\end{array}$

Thus, the value of $\left(f\circ g\right)\left(-2\right)$ is$-3$.

(b)Step 1 – Composition of functions.

Composition of functions is defined as:

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

– Given information.

The given functions are:

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

– Find the required value.

$\begin{array}{c}\left(g\circ f\right)\left(-2\right)=g\left(f\left(-2\right)\right)\\ =g\left(2\left(-2\right)-3\right)\left[\because f\left(x\right)=2x-3\right]\\ =g\left(-4-3\right)\\ =g\left(-7\right)\\ =4-{\left(-7\right)}^2\left[\because g\left(x\right)=4-x^2\right]\\ =4-49\\ =-45\end{array}$

Thus, the value of$\left(g\circ f\right)\left(-2\right)$ is $-45$.