Q. 16 - Precalculus Enhanced with Graphing Utilities

Question

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

$f\left(x\right)=\sqrt{x+1},g\left(x\right)=3x$.

(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) $\sqrt{13}$ (b) $3\sqrt{3}$ (c) $3\sqrt{2}$ (d) 0.

Part (a) Step 1. Given Information.

The given function are :

$f\left(x\right)=\sqrt{x+1},g\left(x\right)=3x$.

Part (a) Step 2. The evaluation of a functional value.

The functional value are :

(a) $\left(f\circ g\right)\left(4\right)=f\left(g\left(4\right)\right)=f\left(\frac{1}{17}\right)=\frac{1}{17}$$\left(f\circ g\right)\left(4\right)=f\left(g\left(4\right)\right)=f\left(12\right)=\sqrt{13}$.

Part (b) Step 1. Given Information.

The given function is 

$f\left(x\right)=\sqrt{x+1},g\left(x\right)=3x$.

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

The value of a function is 

$\left(g\circ f\right)\left(2\right)=g\left(f\left(2\right)\right)=g\left(\sqrt{13}\right)=3\sqrt{3}$.

Part (c) Step 1. Given Information.

The given function is 

$f\left(x\right)=\sqrt{x+1},g\left(x\right)=3x$.

Part (c) Evaluation of a functional value.

The value of the function is $\left(f\circ f\right)\left(1\right)=f\left(f\left(1\right)\right)=f\left(\sqrt{2}\right)=3\sqrt{2}$.

Part (d) Step 1. Given Information.

The given function is 

$f\left(x\right)=\sqrt{x+1},g\left(x\right)=3x$.

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

The value of the function is 

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