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. $\left(f\circ g\right)\left(x\right)$
   
2. $\left(g\circ f\right)\left(x\right)$
   

Short answer

1. The value of$\left(f\circ g\right)\left(x\right)$ is $5-2x^2$.
2. The value of$\left(g\circ f\right)\left(x\right)$ is $-4x^2+12x-5$.

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.

$\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ =f\left(4-x^2\right)\left[\because g\left(x\right)=4-x^2\right]\\ =2\left(4-x^2\right)-3\left[\because f\left(x\right)=2x-3\right]\\ =8-2x^2-3\\ =5-2x^2\end{array}$

Thus, the value of$\left(f\circ g\right)\left(x\right)$ is $5-2x^2$.

(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.

$\begin{array}{c}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\\ =g\left(2x-3\right)\left[\because f\left(x\right)=2x-3\right]\\ =4-{\left(2x-3\right)}^2\left[\because g\left(x\right)=4-x^2\right]\\ =4-\left({\left(2x\right)}^2-2\left(2x\right)\left(3\right)+{\left(3\right)}^2\right)\left[\because {\left(a-b\right)}^2=a^2-2ab+b^2\right]\\ =4-\left(4x^2-12x+9\right)\\ =4-4x^2+12x-9\\ =-4x^2+12x-5\end{array}$

Thus, the value of$\left(g\circ f\right)\left(x\right)$ is $-4x^2+12x-5$.