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Chapter 5: Q. 98 (page 285)

In Problems 97-100,graph each function,Based on the graph,state the domain and the range and find any intercepts.

f(x)=exifx<0e-xifx≥0

Short Answer

Expert verified

Graph of the given function

Domain of the given function (-∞,∞).

Range of the given function (0,∞).

The y-intercept is(0,1) . The two functions have the liney=0 as their horizontal asymptote. As the value ofx increases, exbecomes smaller and the graph will approach the liney=0 . But, it will not intersect the line. Similar is the case with the graph of e-xalso. This implies that there is no x-intercept.

Step by step solution

01

Step 1.Given information

The given functionf(x)=exifx<0e-xifx≥0

02

Step 2.Finding the x and y values for f(x)=ex

Let us begin with the graph of f(x)or y=ex . Since this function is defined for the value of x less than zero, select some values for xthat are less than 0 and find the corresponding values ofy.

Create a table using the values.
x y=ex
-30
-20.1
-10.4
03

Step 3.Finding the x and y values for y=e-x

Similarly, for the function y=e-x, select some values for x that are greater than or equal to 0 .
xy=e-x
01
10.4
20.1
04

Step 4.Plot the points for the two functions. Connect each set of points using a smooth curve. 

The domain of a function is the set of all x-values for which the function is defined. We can see that f(x) is defined for all real numbers. The domain of f(x) is the interval (-∞,∞).

The range is the set of values of the function. From the graph, it is possible to say that the range of f(x) is the interval (0,∞).
The y-intercept is (0,1). The two functions have the line y=0 as their horizontal asymptote. As the value of x increases, exbecomes smaller and the graph will approach the line y=0. But, it will not intersect the line. Similar is the case with the graph ofe-x also. This implies that there is no x-intercept.

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Most popular questions from this chapter

The graphs ofy=a-x and y=1axare identical. Why?

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

True or False

To graph (x-2)3, shift the graph of y=x3to the left 2 units.

Iffx=ax, show thatfx+h-fxh=ax.ah-1hh≠0.

Begin with the graph of y=exand use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

fx=ex-1

In Problems 61–72, the function f is one-to-one. Find its inverse and check your answer.

f(x)=-2xx-1
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