Cost–Benefit Let f(x) be the cost–benefit function from Example. If 70% of the pollutant h...

Cost–Benefit Let f(x) be the cost–benefit function from Example. If 70% of the pollutant has been removed, what is the added cost to remove another 5%? How does this compare with the cost to remove the final 5% of the pollutant? (See Example.)

Example

A Cost–Benefit Model Suppose that a cost–benefit function is given by

f(x) =  0 ≤ x ≤ 100,

where x is the percentage of some pollutant to be removed and f(x) is the associated cost (in millions of dollars). (See Fig. 1.) Find the cost to remove 70%, 95%, and 100% of the pollutant.

Figure 1

SOLUTION

The cost to remove 70% is f(70) =  = 100 (million dollars).

Similar calculations show that

f(95) = 475 and f(100) = 1000.

Observe that the cost to remove the last 5% of the pollutant is f(100) − f(95) = 1000 − 475 = 525 million dollars. This is more than five times the cost to remove the first 70% of the pollutant!

Power Functions Functions of the form f(x) = xr are called power functions. The meaning of xr is obvious when r is a positive integer. However, the power function f(x) = xr may be defined for any number r. We delay until Section 0.5, a discussion of power functions, where we will review the meaning of xr in the case when r is a rational number.

The Absolute Value Function The absolute value of a number x is denoted by |x| and is defined by

|x| =

For example, |5| = 5, |0| = 0, and |−3| = −(−3) = 3.

The function defined for all numbers x by

f(x) = |x|

is called the absolute value function. Its graph coincides with the graph of the equation y = x for x ≥ 0 and with the graph of the equation y = −x for x<0. (See Fig. 1)

Figure 1 Graph of the absolute value function.