Chapter 5.1, Problem 31AYU

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.

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

To determine

To find:

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

State the domain of each composite function.

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

Answer to Problem 31AYU

a. The domain of $\left(f\circ g\right)\left(x\right)=3x^2+1$.

Explanation of Solution

Given:

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

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,

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

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

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

To determine

To find:

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

State the domain of each composite function.

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

Answer to Problem 31AYU

b. The domain of $\left(g\circ f\right)\left(x\right)=9x^2+6x+1$.

Explanation of Solution

Given:

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

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,

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

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

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

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

To determine

To find:

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

State the domain of each composite function.

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

Answer to Problem 31AYU

c. The domain of $\left(f\circ f\right)\left(x\right)=9x+4$.

Explanation of Solution

Given:

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

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,

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

$\left(f\circ f\right)\left(x\right)=f\left(3x+1\right)$

$\left(f\circ f\right)\left(x\right)=3\left(3x+1\right)+1$

$\left(f\circ f\right)\left(x\right)=9x+3+1$

$\left(f\circ f\right)\left(x\right)=9x+4$

To determine

To find:

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

State the domain of each composite function.

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

Answer to Problem 31AYU

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

Explanation of Solution

Given:

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

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,

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

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

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

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