Chapter 2.4, Problem 36AYU

In Problems 31-42:

(a) Find the domain of each function.

(b) Locate any intercepts.

(c) Graph each function.

(d) Based on the graph, find the range.

$f\left(x\right)=\{\begin{array}{l}\frac{1}{x}\text{if}x<0\\ \sqrt[3]{x}\text{if}x\ge 0\end{array}$

To determine

To find: The following values of the function,$f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$:

a. Domain of the function $f$.

Answer to Problem 36AYU

Solution:

a. Domain of the function $f$: The set of all real numbers. That is, $\left(-\infty ,\infty \right)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$

a. Domain of the function $f$: To find the domain of $f$, look at its definition. Since $f$ is defined for all $x<0$ and $x\ge 0$, the domain of $f$ is $\left(-\infty ,\infty \right)$.

To determine

To find: The following values of the function,$f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$:

b. Intercepts of the function $f$ if any.

Answer to Problem 36AYU

Solution:

b. Intercepts of the function $f$ if any: $x\text{-intercept is }(0,0)\text{ and }y\text{-intercept is also }(0,0)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$

b. Intercepts: The $y\text{-intercept}$ of the graph of the function is $f\left(0\right)$. Because the equation for $f$ when $x=0$ is $f\left(x\right)=\sqrt[3]{x}$, the $y\text{-intercept}$ is $y=\sqrt[3]{0}=0$.

The $x\text{-intercepts}$ of the graph of a function $f$ are the real solutions to the equation $f\left(x\right)=0$. To find the $x\text{-intercepts}$ of $f$, solve $f\left(x\right)=0$ for each “piece” of the function, and then determine which values of x, if any, satisfy the condition that defines the piece.

$f\left(x\right)=0$

$\frac{1}{x}=0$

$x=\infty$

Contradicts the condition $x<0$

Satisfies the condition $x\ge 0$.

Therefore, the $x\text{-intercept is }(0,0)$.

To determine

To find: The following values of the function,$f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$:

c. Graph of the function $f$.

Answer to Problem 36AYU

Solution:

c. Graph of the function $f$:

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$

c.

To determine

To find: The following values of the function,$f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$:

d. Range of the function $f$ based on its graph.

Answer to Problem 36AYU

Solution:

d. Range of the function $f$ based on its graph: The range takes the set of all real values. $\text{Range}=\left(-\infty ,\infty \right)$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}\frac{1}{x} if x<0\\ \sqrt[3]{x} if x\ge 0\end{array}$

d. From the graph, it can be easily predicted that the takes the set of all real values. Range $=\left(-\infty ,\infty \right)$