Chapter 7.4, Problem 2MCQ

To determine

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

Answer to Problem 2MCQ

The domain is $\left\{x:x\text{ is a real number}\right\}$ and the range is $\left\{y:y<5\right\}$ .

Explanation of Solution

Given:

The given function is:

  $f\left(x\right)=-{\left(2\right)}^x+5\dots \left(1\right)$

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

  $\begin{array}{c}\underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }-{\left(2\right)}^x+5\\ =-{\left(2\right)}^{-\infty }+5\\ =-\infty \end{array}$

Further simplified as:

  $\begin{array}{c}\underset{x\to +\infty }{\lim }f\left(x\right)=\underset{x\to +\infty }{\lim }-{\left(2\right)}^x+5\\ =-\infty \end{array}$

Therefore by using the graph below and above explanation the range is $\left\{y:y<5\right\}$ .

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

$x$	$0$	$1$	$-1$
$f\left(x\right)$	$4$	$3$	$\frac{9}{2}$

The graph of the given function is given by: