Chapter 1.2, Problem 20E

(a)

To determine

To find:The domain and range of the function $y=\frac{1}{\sqrt{4-x^2}}$ .

Answer to Problem 20E

The domain and range of the function $y=\frac{1}{\sqrt{4-x^2}}$ is $\left(-2,2\right)$ and $\left[0.5,\infty \right)$ respectively.

Explanation of Solution

Given information:The given function is $y=\frac{1}{\sqrt{4-x^2}}$ .

Calculation:

Consider the given function.

  $y=\frac{1}{\sqrt{4-x^2}}$

Domain is the set of all input values. The given function is defined only for $x$ between $-2$ to $2$ .So, the domain of the function is the interval $\left(-2,2\right)$ .

Range is the set of output values. It can be observed that the value of $y$ is always greater than $0.5$ . So, the range of the functionis $\left[0.5,\infty \right)$ .

Therefore, the domain and range of the function $y=\frac{1}{\sqrt{4-x^2}}$ is $\left(-2,2\right)$ and $\left[0.5,\infty \right)$ respectively.

(b)

To determine

To graph:The function $y=\frac{1}{\sqrt{4-x^2}}$ with the help of graphing calculator.

Explanation of Solution

Given information:The function is $y=\frac{1}{\sqrt{4-x^2}}$ .

Graph:

To graph a function $f\left(x\right)=\frac{1}{\sqrt{4-x^2}}$ , 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 $f\left(x\right)=\frac{1}{\sqrt{4-x^2}}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{(}\boxed{4}\boxed{-}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{)}\boxed{^}\boxed{(}\boxed{1}\boxed{÷}\boxed{2}\boxed{)}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=1/\left(\left(4-\text{X}^2\right)^\left(1/2\right)\right)$

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 $y=\frac{1}{\sqrt{4-x^2}}$ is curve with no $x$ -intercept and $0.5$ as $y$ -intercept.