Q. 15 - Precalculus Enhanced with Graphing Utilities

Question

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

$f\left(x\right)=\sqrt{x},g\left(x\right)=2x$.

(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) $2\sqrt{2}$ (b) $2\sqrt{2}$ (c) 1 (d) 0.

Part (a) Step 1. Given Information.

The given functions are :

$f\left(x\right)=\sqrt{x},g\left(x\right)=2x$.

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

The functional value is :

(a) $\left(f\circ g\right)\left(4\right)=f\left(g\left(4\right)\right)=f\left(8\right)=2\sqrt{2}$.

Part (b) Step 1. Given Information.

The given function be 

$f\left(x\right)=\sqrt{x},g\left(x\right)=2x$.

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

The value of the function is 

$\left(g\circ f\right)\left(2\right)=g\left(f\left(2\right)\right)=g\left(\sqrt{2}\right)=2\sqrt{2}$.

Part (c) Step 1. Given Information.

The given function is 

$f\left(x\right)=\sqrt{x},g\left(x\right)=2x$.

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

The value of the function is 

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

Part (d) Step 1. Given Information.

The given function be

$f\left(x\right)=\sqrt{x},g\left(x\right)=2x$.

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

The value of the function be 

$\left(g\circ g\right)\left(0\right)=g\left(g\left(0\right)\right)=g\left(0\right)=0$.