Chapter 2.2, Problem 45AYU

To determine

To draw: a graph of a function with given domain and range. The point on the rectangle which does not lie on the graph.

Answer to Problem 45AYU

The graph of this function is the single point $\left(1,1\right)$ and its equation is $1=1$ .

Explanation of Solution

Given information:

A function whose domain is $\left\{x|-3\le x\le 8,x\ne 5\right\}$ and whose range is $\left\{y|-1\le y\le 2,y\ne 0\right\}$ . The rectangle $-3\le x\le 8,-1\le y\le 2$ .

Calculation:

Consider the function

  $\begin{array}{c}f\left(x\right)=-\frac{1}{\frac{10\sqrt{\left(x+3\right)\left(8-x\right)}}{x-5}}\\ =-\frac{x-5}{10\sqrt{\left(x+3\right)\left(8-x\right)}},\left(-3<x<8\right)\cap \left(x\ne 5\right)\\ =-\frac{x-5}{10\sqrt{24+5x-x^2}},\left(-3<x<8\right)\cap \left(x\ne 5\right)\end{array}$

The domain of this function $\left\{x|-3\le x\le 8,x\ne 5\right\}$ . The range of this function is $\left\{y|-1\le y\le 2,y\ne 0\right\}$ . We can plot this function using Maple as given below. The function has a discontinuity at $\left(5,0\right)$ as $x\ne 5$ .

  

No points whose $x$ coordinate is 5 or whose $y$ coordinate is 0 on the graph. Different graphs satisfying the given domain and range may differ in shape.