Chapter 2.4, Problem 31AYU

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.

$f\left(x\right)=\{\begin{array}{l}-2x+3\text{if}x<1\\ 3x-2\text{if}x\ge 1\end{array}$

To determine

To find: The following values of the function $f\left(x\right)=\{\begin{array}{c}-2x+3 if x<1\\ 3x-2 if x\ge 1\end{array}$

a. Domain of the function $f$.

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

c. Graph of the function $f$.

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

Answer to Problem 31AYU

Solution:

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

b. Intercepts of the function $f$ if any: $y\text{-intercept is }\left(0,3\right). \text{There is no }x\text{-intercept}$.

c. Graph of the function $f$:

d. Range of the function $f$ based on its graph: The range takes the real values which are greater than or equal to 1. $\text{Range }=\left\{x/x\ge 1,x\in R\right\}$.

Explanation of Solution

Given:

The function is: $f\left(x\right)=\{\begin{array}{c}-2x+3 if x<1\\ 3x-2 if x\ge 1\end{array}$

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 $x<1$ and $x\ge 1$, the domain of $f$ is the set of all real numbers. That is, $\left(-\infty ,\infty \right)$.

b. Intercepts: The $y\text{-intercept}$ of the graph of the function is $f(0)$. Because the equation for $f$ when $x=0$ is $f\left(x\right)=-2X+3$, the $y\text{-intercept}$ is $f\left(0\right)=-2\left(0\right)+3=3$.

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

$f\left(x\right)=\text{ 0}$

$-2x+3=0$

$x=\frac{3}{2}$

Contradicts the condition $x<1$.

$f\left(x\right)=\text{ 0}$

$3x-2=0$

$x=\frac{2}{3}$

Contradicts the condition $x\ge 1$.

There is no $x\text{-intercept}$ for the given graph.

c.

d. From the graph, it can be easily predicted that the range takes all the real values which are greater than or equal to 1. $\text{Range }=\left\{x/x\ge 1,x\in R\right\}$