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. $f\left(g\left(0\right)\right)$
   
2. $g\left(f\left(0\right)\right)$
   

Short answer

1. The value of $f\left(g\left(0\right)\right)$is 5 .
2. The value of $g\left(f\left(0\right)\right)$is -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.

$\begin{array}{c}f\left(g\left(0\right)\right)=f\left(4-{\left(0\right)}^2\right)\left[\because g\left(x\right)=4-x^2\right]\\ =f\left(4-0\right)\\ =f\left(4\right)\\ =2\left(4\right)-3\left[\because f\left(x\right)=2x-3\right]\\ =8-3\\ =5\end{array}$

Thus, the value of$f\left(g\left(0\right)\right)$ is 5.

(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.

$\begin{array}{c}g\left(f\left(0\right)\right)=g\left(2\left(0\right)-3\right)\left[\because f\left(x\right)=2x-3\right]\\ =g\left(0-3\right)\\ =g\left(-3\right)\\ =4-{\left(-3\right)}^2\left[\because g\left(x\right)=4-x^2\right]\\ =4-9\\ =-5\end{array}$

Thus, the value of$g\left(f\left(0\right)\right)$ is $-5$.