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Chapter 2: Q 48. (page 90)

In Problems 45–52, for each graph of a function y=f(x), find the absolute maximum and the absolute minimum, if they exist.

Short Answer

Expert verified

The absolute minimum is 1 and there is no absolute maximum.

Step by step solution

01

Step 1. Given information.

The given graph of the function y=f(x)is:

02

Step 2. Use the concept of absolute maximum and absolute minimum.

Let fdenote a function defined on some interval I. If there is a number u in I for which f(x)≤f(u) for all x in I, then f(u)is the absolute maximum of fand I.

If there is a number v in I for which f(x)≥f(v) for all x in I , then f(v)is the absolute minimum of fonI.

03

Step 3. Find the absolute maximum. 

We can see from the graph that the given function has the domain {x0≤x<3,x≠2}.

Here we are excluding 2 from the domain because of the "hole" at (2,4).

There is no absolute maximum.

The reason being, as we trace the graph getting closer to the point (2,4), there is no single largest value.

04

Step 4. Find the absolute minimum.

We can see from the graph that the given function has a minimum value at its domain is:

f(0)=1

The smallest value of fisf(0)=1.

Therefore, the absolute minimum of the function is 1.

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Most popular questions from this chapter

If f(x)=-3xifx<-10ifx=-12x2+1ifx>-1

Find: (a)f(-2) and (b) f(-1) (c)f(0)

True or False The independent variable is sometimes

referred to as the argument of the function.

In Problems 7–18, match each graph to one of the following functions:

A.y=x2+2B.y=-x2+2C.y=x+2D.y=-x+2E.y=(x-2)2F.y=-(x+2)2G.y=x-2H.y=-x+2I.y=2x2J.y=-2x2K.y=2xL.y=-2x

A semicircle of radius r is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle. See the figure.

(a) Express the area A of the rectangle as a function of the radius r of the semicircle.

(b) Express the perimeter p of the rectangle as a function of r.

Graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. Verify your results using a graphing utility.

f(x)=-x
See all solutions

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