Chapter 2.4, Problem 35AYU

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.

$f\left(x\right)=\{\begin{array}{l}1+x\text{if}x<0\\ x^2\text{if}x\ge 0\end{array}$

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

a. Domain of the function $f$.

Answer to Problem 35AYU

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: $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

It is asked to find the domain and intercepts of the function $f$. Also, sketch the function and conclude its range from it.

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 $x=0$, the domain of $f$ is $\left(-\infty ,\infty \right)$.

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

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

Answer to Problem 35AYU

Solution:

b. Intercepts of the function $f$ if any: $x\text{-intercepts are }(-1,0)\text{ and }(0,0)\text{ and }y\text{-intercept is }(0,0)$.

Explanation of Solution

Given: The function is: $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

It is asked to find the domain and intercepts of the function $f$. Also, sketch the function and conclude its range from it.

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 $f\left(x\right)=x^2$, the $y\text{-intercept}$ is $y=0^2=0$.

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.

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

$1+x=0$

$x=-1$

Satisfies the condition $x<0$

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

$x^2=0$

Satisfies the condition $x\ge 0$.

Therefore, the $x\text{-intercepts are }(-1,0)\text{ and }(0,0)$.

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

c. Graph of the function $f$.

Answer to Problem 35AYU

Solution:

c. Graph of the function $f$:

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

It is asked to find the domain and intercepts of the function $f$. Also, sketch the function and conclude its range from it.

c.

To determine

To find: The following values of the function, $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

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

Answer to Problem 35AYU

Solution:

d. Graph of the function $f$:

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}1+x if x<0\\ x^2 if x\ge 0\end{array}$

It is asked to find the domain and intercepts of the function $f$. Also, sketch the function and conclude its range from it.

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