Chapter 5.1, Problem 34AYU

In Problems 23-38, for the given functions $f$ and $g$, find:

a. $f\circ g$

b. $g\circ f$

c. $f\circ f$

d. $g\circ g$

State the domain of each composite function.

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

To determine

To find:

a. $f\circ g$ for the given functions $f$ and $g$ ,

State the domain of each composite function.

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

Answer to Problem 34AYU

Solution:

a. The domain of $\left(f\circ g\right)\left(x\right)=4x^4+12x^2+10$.

Explanation of Solution

Given:

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

Calculation:

The domain of $f$ and the domain of $g$ are the set of all real numbers.

Because the domains of both $f$ and $g$ are the set of all real numbers, the domain of $f\circ g$ is the set of all real numbers.

We know the composite function of $f$ and $g$ is defined as

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

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

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

$\left(f\circ g\right)\left(x\right)=\left(4x^4+12x^2+9\right)+1$

$\left(f\circ g\right)\left(x\right)=4x^4+12x^2+10$

To determine

To find:

b. $g\circ f$ for the given functions $f$ and $g$ ,

State the domain of each composite function.

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

Answer to Problem 34AYU

Solution:

b. The domain of $\left(g\circ f\right)\left(x\right)=2x^4+4x^2+5$.

Explanation of Solution

Given:

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

Calculation:

The domain of $f$ and the domain of $g$ are the set of all real numbers.

Because the domains of both $f$ and $g$ are the set of all real numbers, the domain of $f\circ g$ is the set of all real numbers.

We know the composite function of $f$ and $g$ is defined as

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

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

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

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

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

$\left(g\circ f\right)\left(x\right)=2x^4+4x^2+5$

To determine

To find:

c. $f\circ f$ for the given functions $f$ and $g$,

State the domain of each composite function.

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

Answer to Problem 34AYU

Solution:

c. The domain of $\left(f\circ f\right)\left(x\right)=2x^4+2x^2+2$.

Explanation of Solution

Given:

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

Calculation:

The domain of $f$ and the domain of $g$ are the set of all real numbers.

Because the domains of both $f$ and $g$ are the set of all real numbers, the domain of $f\circ g$ is the set of all real numbers.

We know the composite function of $f$ and $g$ is defined as

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

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

$\left(f\circ f\right)\left(x\right)=f\left(x^2+1\right)$

$\left(f\circ f\right)\left(x\right)={\left(x^2+1\right)}^2+1$

$\left(f\circ f\right)\left(x\right)=x^4+2x^2+1+1$

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

To determine

To find:

d. $g\circ g$ for the given functions $f$ and $g$ ,

State the domain of each composite function.

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

Answer to Problem 34AYU

Solution:

d. The domain of $\left(g\circ g\right)\left(x\right)=4x^4+12x^2+12$.

Explanation of Solution

Given:

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

Calculation:

The domain of $f$ and the domain of $g$ are the set of all real numbers.

Because the domains of both $f$ and $g$ are the set of all real numbers, the domain of $f\circ g$ is the set of all real numbers.

We know the composite function of $f$ and $g$ is defined as

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

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

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

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

$\left(g\circ g\right)\left(x\right)=\left(4x^4+12x^2+9\right)+3$

$\left(g\circ g\right)\left(x\right)=4x^4+12x^2+9+3$

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