Chapter 5.1, Problem 14AYU

In Problems 13-22, for the given functions $f$ and $g$, 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. $\left(g\circ g\right)\left(0\right)$

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

To determine

To find: For the given functions $f$ and $g$,

a. $(f\circ g)\left(4\right)$

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Answer to Problem 14AYU

Solution:

a. $\left(f\circ g\right)\left(4\right)=4418$

Explanation of Solution

Given:

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Calculation:

We know the composition of $f$ and $g$ is defined as

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

$\left(f\circ g\right)\left(x\right)=f\left(1-3x^2\right)$

$\left(f\circ g\right)\left(x\right)=2{\left(1-3x^2\right)}^2$

$\left(f\circ g\right)\left(x\right)=2\left(1-6x^2+9x^4\right)$

$\left(f\circ g\right)\left(x\right)=2-12x^2+18x^4$

Here $x=4$

$\left(f\circ g\right)\left(x\right)=2-12{\left(4\right)}^2+18{\left(4\right)}^4$

$\left(f\circ g\right)\left(4\right)=2-192+4608$

$\left(f\circ g\right)\left(4\right)=4418$

To determine

To find: For the given functions $f$ and $g$,

b. $(g\circ f)\left(2\right)$

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Answer to Problem 14AYU

Solution:

b. $\left(g\circ f\right)\left(2\right)=-191$

Explanation of Solution

Given:

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Calculation:

We know the composition of $f$ and $g$ is defined as

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

$\left(g\circ f\right)\left(x\right)=g\left(2x^2\right)$

$\left(g\circ f\right)\left(x\right)=1-3{\left(2x^2\right)}^2$

$\left(g\circ f\right)\left(x\right)=1-3\left(4x^4\right)$

$\left(g\circ f\right)\left(x\right)=1-12x^4$

Here $x=2$

$\left(g\circ f\right)\left(2\right)=1-12{\left(2\right)}^4$

$\left(g\circ f\right)\left(2\right)=1-192$

$\left(g\circ f\right)\left(2\right)=-191$

To determine

To find: For the given functions $f$ and $g$,

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

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Answer to Problem 14AYU

Solution:

c. $\left(f\circ f\right)\left(1\right)=8$

Explanation of Solution

Given:

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Calculation:

We know the composition of $f$ and $g$ is defined as

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

$\left(f\circ f\right)\left(x\right)=f\left(2x^2\right)$

$\left(f\circ f\right)\left(x\right)=2{\left(2x^2\right)}^2$

$\left(f\circ f\right)\left(x\right)=2\left(4x^4\right)$

$\left(f\circ f\right)\left(x\right)=8x^4$

Here $x=1$

$\left(f\circ f\right)\left(1\right)=8{\left(1\right)}^4$

$\left(f\circ f\right)\left(1\right)=8$

To determine

To find: For the given functions $f$ and $g$,

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

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Answer to Problem 14AYU

Solution:

d. $\left(g\circ g\right)\left(0\right)=-2$

Explanation of Solution

Given:

$f\left(x\right)=2x^2$; $g\left(x\right)=1-3x^2$

Calculation:

We know the composition of $f$ and $g$ is defined as

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

$\left(g\circ g\right)\left(x\right)=g\left(1-3x^2\right)$

$\left(g\circ g\right)\left(x\right)=1-3{\left(1-3x^2\right)}^2$

$\left(g\circ g\right)\left(x\right)=1-3\left(1-6x^2+9x^4\right)$

$\left(g\circ g\right)\left(x\right)=1-\left(3-18x^2+27x^4\right)$

$\left(g \circ g\right)\left(x\right) = -2 + 18x^2 - 27x^4$

Here $x=0$

$\left(g\circ g\right)\left(0\right)=-2+18{\left(0\right)}^2-27{\left(0\right)}^4$

$\left(g\circ g\right)\left(0\right)=-2+0-0$

$\left(g\circ g\right)\left(0\right)=-2$