Chapter 5.1, Problem 33AYU

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.

$f(x)= x^{2 }$; $g(x)= x^2+4$

To determine

To find:

a. $f \circ g$ for the given functions $f$ and $g$, State the domain of each composite function.

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

Answer to Problem 33AYU

a. The domain of $\left(f \circ g\right)\left(x\right) = x^4 + 4x^2 + 16$

Explanation of Solution

Given:

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

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

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

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

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

$\left(f \circ g\right)\left(x\right) = x^4 + 4x^2 + 16$

To determine

To find:

b. $g \circ f$ for the given functions $f$ and $g$, State the domain of each composite function.

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

Answer to Problem 33AYU

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

Explanation of Solution

Given:

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

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

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

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

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

To determine

To find:

c. $f \circ f$ for the given functions $f$ and $g$, State the domain of each composite function.

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

Answer to Problem 33AYU

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

Explanation of Solution

Given:

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

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

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

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

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

To determine

To find:

d. $g \circ g$ for the given functions $f$ and $g$, State the domain of each composite function.

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

Answer to Problem 33AYU

d). The domain of $\left(g \circ g\right)\left(x\right) = x^4 + 8x^2 + 20$

Explanation of Solution

Given:

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

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

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

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

$\left(g \circ g\right)\left(x\right) = x^4 + 8x^2 + 16 + 4$

$\left(g \circ g\right)\left(x\right) = x^4 + 8x^2 + 20$