Chapter 1.2, Problem 16E

(a)

To determine

To find:The domain and range of the function $y=\sqrt{9-x^2}$ .

Answer to Problem 16E

The domain and range of the function $y=\sqrt{9-x^2}$ is $\left[-3,3\right]$ and $\left[0,3\right]$ respectively.

Explanation of Solution

Given information:The given function is $y=\sqrt{9-x^2}$ .

Calculation:

Consider the given function.

  $y=\sqrt{9-x^2}$

Domain is the set of all input values. The function $y=\sqrt{9-x^2}$ is defined only for $x$ between $-3$ to $3$ . So, the domain of the function is the interval $\left[-3,3\right]$ .

Range is the set of all output values for which function is defined. The output values of the function $y$ always lie between $0$ to $3$ . So, the range of the function is the interval $\left[0,3\right]$ .

Therefore, the domain and range of the function $y=\sqrt{9-x^2}$ is $\left[-3,3\right]$ and $\left[0,3\right]$ respectively.

(b)

To determine

To graph:The function $y=\sqrt{9-x^2}$ with the help of graphing calculator.

Explanation of Solution

Given information:The function is $y=\sqrt{9-x^2}$ .

Graph:

To graph a function $f\left(x\right)=\sqrt{9-x^2}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(x\right)=\sqrt{9-x^2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{9}\boxed{-}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{)}\boxed{^}\boxed{(}\boxed{1}\boxed{÷}\boxed{2}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(9-\text{X}^2\right)^\left(1/2\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown below:

  

Figure (1)

Interpretation:The graph of the function $y=\sqrt{9-x^2}$ is a semicircle with center at origin and radius $3$ units.