Question

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

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

Short answer

1. The value of$\left(f\circ f\right)\left(-1\right)$ is $-13$.
2. The value of$\left(g\circ g\right)\left(-1\right)$ is $-5$.

Step-by-step solution

(a)Step 1 – Composition of functions.

Composition of functions is defined as:

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

– 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.

$\begin{array}{c}\left(f\circ f\right)\left(-1\right)=f\left(f\left(-1\right)\right)\\ =f\left(2\left(-1\right)-3\right)\left[\because f\left(x\right)=2x-3\right]\\ =f\left(-2-3\right)\\ =f\left(-5\right)\\ =2\left(-5\right)-3\left[\because f\left(x\right)=2x-3\right]\\ =-10-3\\ =-13\end{array}$

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

(b)Step 1 – Composition of functions.

Composition of functions is defined as:

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

– 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.

$\begin{array}{c}\left(g\circ g\right)\left(-1\right)=g\left(g\left(-1\right)\right)\\ =g\left(4-{\left(-1\right)}^2\right)\left[\because g\left(x\right)=4-x^2\right]\\ =g\left(4-1\right)\\ =g\left(3\right)\\ =4-{\left(3\right)}^2\left[\because g\left(x\right)=4-x^2\right]\\ =4-9\\ =-5\end{array}$

Thus, the value of$\left(g\circ g\right)\left(-1\right)$ is $-5$.