Chapter 8, Problem 4MCQ

To determine

To find: The domain and the range of given function and plot the graph.

Answer to Problem 4MCQ

The domain is $\left\{x:x\text{ is a real number}\right\}$ and the range is $\left\{y:y<8\right\}$ .

Explanation of Solution

Given:

The given function is:

  $f\left(x\right)=-3{\left(\frac{2}{3}\right)}^{x-1}+8\dots \left(1\right)$

The domain is $\left\{x:x\text{ is a real number}\right\}$ .

For range further simplified as:

  $\begin{array}{c}\underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }-3{\left(\frac{2}{3}\right)}^{x-1}+8\\ =-3{\left(\frac{2}{3}\right)}^{-\infty -1}+8\\ =-\infty \end{array}$

Further simplified as:

  $\begin{array}{c}\underset{x\to +\infty }{\lim }f\left(x\right)=\underset{x\to +\infty }{\lim }-3{\left(\frac{2}{3}\right)}^{x-1}+8\\ =-\infty \end{array}$

Therefore by using the graph below and above explanation the range is $\left\{y:y<8\right\}$ .

The table for the graph of the function is given as:

$x$	$0$	$1$	$-2$
$f\left(x\right)$	$3.5$	$\frac{5}{4}$	$\frac{17}{8}$

The graph of the given function is given by: