Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

Find for the given functions$f,g$ 

$f\left(x\right)=4x^2-3,g\left(x\right)=3-\frac{1}{2}{x}^2$.

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) 97 (b) $\frac{-163}{2}$ (c) 1 (d) $-\frac{3}{2}$.

Part (a) Step 1. Given Information.

The given functions are

 $f\left(x\right)=4x^2-3,g\left(x\right)=3-\frac{1}{2}{x}^2$.

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

The functional valuee is : 

(a) $\left(f\circ g\right)\left(4\right)=f\left(g\left(4\right)\right)=f\left(-5\right)=100-3=97$.

Part (b) Step 1. Given Information.

The given function is 

$f\left(x\right)=4x^2-3,g\left(x\right)=3-\frac{1}{2}{x}^2$.

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

The value of a function is 

$\left(g\circ f\right)\left(2\right)=g\left(f\left(2\right)\right)=g\left(13\right)=3-\frac{169}{2}=-\frac{163}{2}$.

Part (c) Step 1. Given Information.

The value of the function is 

$f\left(x\right)=4x^2-3,g\left(x\right)=3-\frac{1}{2}{x}^2$.

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 Infoormation.

The given function is 

$f\left(x\right)=4x^2-3,g\left(x\right)=3-\frac{1}{2}{x}^2$.

Step 2. Evaluation of a functional value.

The value of the function is 

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