Chapter 2.3, Problem 10QR

To determine

To find:The value of $c$ such that $f\left(x\right)=c$ has no solution for $f\left(x\right)=\{\begin{array}{l}5-x,x\le 3\\ -x^2+6x-8,x>3\end{array}$ .

Answer to Problem 10QR

Any value of $c$ does not exist such that equation $f\left(x\right)=c$ has no solution.

Explanation of Solution

Given information:The functionis $f\left(x\right)=\{\begin{array}{l}5-x,x\le 3\\ -x^2+6x-8,x>3\end{array}$ .

Calculation:

To graph a function $f\left(x\right)=\{\begin{array}{l}5-x,x\le 3\\ -x^2+6x-8,x>3\end{array}$ , 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 $y=\{\begin{array}{l}5-x,x\le 3\\ -x^2+6x-8,x>3\end{array}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{5}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{6}\boxed{3}\boxed{)}$ . Now press Enter after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{(}\boxed{(-)}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{+}\boxed{6}\boxed{\text{X}}\boxed{-}\boxed{8}\boxed{)}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{3}\boxed{)}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\left(5-\text{X}\right)/\left(\text{X}\le 3\right)\\ {\text{Y}}_2=\left(-\text{X}^2+6\text{X}-8\right)/\left(\text{X}>3\right)\end{array}$

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

  

Figure(1)

As observed from graph, the function $y=f\left(x\right)$ is defined for all the values of $x$ .

Therefore, any value of $c$ does not exist such that equation $f\left(x\right)=c$ has no solution.