Chapter 1.2, Problem 9QR

Solution

To find: all the values of $x$ algebraically for which the expression is not defined and show it graphically.

  $x<-2,x\ge 3$

Given Information:

Consider the function $\frac{\sqrt{x+2}}{\sqrt{3-x}}$ .

Concept used:

Consider the following three cases:

• The square root in the numerator has a negative radicand.
• The square root in the denominator has a negative radicand.
• The denominator equals to zero.

Calculation:

Case 1. Suppose that $x+2<0$

  $\begin{array}{c}x+2<0\\ x<-2\end{array}$

Case 2. Suppose that $3-x<0$

  $\begin{array}{c}3-x<0\\ x>3\end{array}$

Case 3. Suppose that $\sqrt{3-x}=0$

  $\begin{array}{c}\sqrt{3-x}=0\\ x=3\end{array}$

Therefore, there are three conditions for the expression to be undefined. Combine the second and third condition $x\ge 3$ .

Therefore, the algebraic condition will be undefined if $x<-2,x\ge 3$ .

Plot the graph to show the expression is not defined graphically also.

  

It is observed from the graph that the graph has no points plotted at $x<-2,x\ge 3$ .

Therefore, the algebraic expression and the graph is not defined at $x<-2,x\ge 3$ .