Chapter 1.3, Problem 19E

(a)

To determine

To calculate: domain of the function $f(x)=\left|x-1\right|-2$ .

Answer to Problem 19E

Domain is $[-2,\infty )$ .

Explanation of Solution

Given information: graph of a function.

Calculation:

Domain of the function $f(x)$ refers to the values of $x$ for which the function is defined.

Graphically, the domain consists of all values of the input shown on x-axis.

The graph is given as follows:

  

According to the given graph, domain is $[-2,\infty )$ as the input values on the graph are $-2\le x<\infty$

(b)

To determine

To calculate: range of the function $f(x)=\left|x-1\right|-2$

Answer to Problem 19E

Range is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Calculation:

Range of the function $f(x)$ refers to its values for the values of $x$ in domain.

Graphically, the range is a set of possible output values shown on y-axis.

The graph is given as follows:

  

According to the given graph, range is $\left(-\infty ,\infty \right)$ as the output values on y-axis is a set of all real numbers.

(c)

To determine

To calculate: values of $x$ for which $f(x)=0$

Answer to Problem 19E

Value of $x$ are $-1,3$ .

Explanation of Solution

Calculation:

The values of $x$ for which $f(x)=0$ refers to the x-coordinate of the points where the graph cuts the x-axis.

The graph is given as follows:

  

The graph cuts the x-axis at points $\left(-1,0\right),\text{}\left(3,0\right)$ .

x -coordinate of point $\left(-1,0\right)$ is $-1$ .

x -coordinate of point $\left(3,0\right)$ is $3$ .

So, values of x for which $f(x)=0$ are $-1,\text{}3$ .

(d)

To determine

To calculate: value of $f(0)$ .

Answer to Problem 19E

  $f(0)=-1$ .

Explanation of Solution

Calculation:

To find the value of $f(0)$ , find the y -coordinate of the point where the graph cuts the y -axis.

The graph is given as follows:

  

The graph cuts the y-axis at point $\left(0,-1\right)$ .

y -coordinate of point $\left(0,-1\right)$ is $-1$ .

So,

  $f(0)=-1$ .

(e)

To determine

To calculate: value of $f(1)$ and coordinates of the point.

Answer to Problem 19E

  $f(1)=-2$ and coordinates of the point are $(1,-2)$ .

Explanation of Solution

Calculation:

The graph is given as follows:

  

Observe the point marked as A in the graph.

Coordinates of point A $=(1,-2)$

  $f(1)$ refers to the y-coordinate of the point A.

So,

  $f(1)=-2$ .

(f)

To determine

To calculate: coordinates of the point on the graph of f at which $x=-3$ .

Answer to Problem 19E

Coordinates of the point on the graph of f at which $x=-3$ are $(-3,2)$

Explanation of Solution

Calculation:

The graph is given as follows:

  

The equation of the given graph is $f(x)=|x-1|-2$ .

To find y -coordinate of the point on the graph of f at which $x=-3$ , put $x=-3$ in $f(x)=|x-1|-2$ .

  $f(-3)=|-3-1|-2=|-4|-2=4-2=2$ .

So, coordinates of the point on the graph of f at which $x=-3$ are $(-3,2)$ .

Hence,

The coordinates of the point on the graph of f at which $x=-3$ can be labelled as $\left(-3,f(-3)\right)$ or $(-3,2)$ .