Q 18. - Precalculus Enhanced with Graphing Utilities

Question

Find for the given functions$f,g$ such that

$f\left(x\right)=\left|x-2\right|, g\left(x\right)=\frac{3}{x^2+2}$.

Find : (a)  $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Step-by-Step Solution

Answer

The value is (a) $\frac{11}{6}$ (b) $\frac{3}{2}$ (c) 1 (d) $\frac{12}{17}$.

Part (a) Step 1. Given Information.

The given functions are : 

$f\left(x\right)=\left|x-2\right| , g\left(x\right)=\frac{3}{x^2+2}$.

Part (a) Step 2. Evaluation of a functional value.

The functional value is :

(a) $\left(f\circ g\right)\left(4\right)=f\left(g\left(4\right)\right)=f\left(\frac{1}{6}\right)=\left(\left|\frac{1}{6}-2\right|\right)=\frac{11}{6}$.

Part (b) Step 1. Given Information.

The given function be 

$f\left(x\right)=\left|x-2\right| , g\left(x\right)=\frac{3}{x^2+2}$.

Part (b) Step 2. Evaluation of a functional value.

The value of the function be 

$\left(g\circ f\right)\left(2\right)=g\left(f\left(2\right)\right)=g\left(0\right)=\frac{3}{2}$.

Part (c) Step 1. Given Information.

The given function be 

$f\left(x\right)=\left|x-2\right| , g\left(x\right)=\frac{3}{x^2+2}$.

Part (c) Evaluation of a functional value.

The value of the function be 

$\left(f\circ f\right)\left(1\right)=f\left(f\left(1\right)\right)=f\left(1\right)=1$.

Part (d) Given Information.

The given function be

$f\left(x\right)=\left|x-2\right| , g\left(x\right)=\frac{3}{x^2+2}$.

Part (d) Evaluation of a functional value.

The value of the function be

$\left(g\circ g\right)\left(0\right)=g\left(g\left(0\right)\right)=g\left(\frac{3}{2}\right)=\frac{3}{2+{\displaystyle \frac{9}{4}}}=\frac{12}{17}$.