Find each function value. See Example 9 . The function f(x)=(100,000 x)/(100-x) models the cost

Find each function value. See Example 9 . The function...

Find each function value. See Example $9 .$ The function $f(x)=\frac{100,000 x}{100-x}$ models the cost in dollars for removing $x$ percent of the pollutants from a bayou in which a nearby company dumped creosol. a. Find the cost of removing $20 \%$ of the pollutants from the bayou. [Hint: Find $f(20) .]$ b. Find the cost of removing $60 \%$ of the pollutants and then $80 \%$ of the pollutants. c. Find $f(90),$ then $f(95),$ and then $f(99) .$ What happens to the cost as $x$ approaches $100 \% ?$ d. Find the domain of function $f$

Find each function value. See Example $9 .$
The function $f(x)=\frac{100,000 x}{100-x}$ models the cost in dollars for removing $x$ percent of the pollutants from a bayou in which
a nearby company dumped creosol.
a. Find the cost of removing $20 \%$ of the pollutants from the bayou. [Hint: Find $f(20) .]$
b. Find the cost of removing $60 \%$ of the pollutants and then $80 \%$ of the pollutants.
c. Find $f(90),$ then $f(95),$ and then $f(99) .$ What happens to the cost as $x$ approaches $100 \% ?$
d. Find the domain of function $f$

Key Concepts

Limits and Asymptotic Behavior

Limits describe the behavior of a function as its input approaches a particular value, and they are especially important for rational functions. When the input approaches a value that makes the denominator zero, the function may approach infinity or negative infinity, indicating the existence of a vertical asymptote. This concept helps in understanding how the function behaves near critical points.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, determining the domain involves identifying and excluding values that cause the denominator to be zero. This ensures that every input within the domain yields a valid output.

Function Evaluation

Function evaluation involves substituting a specific input into a function to calculate the corresponding output. This process is essential in understanding how changes in the independent variable affect the dependent variable and is often used to analyze real-world phenomena modeled by the function.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. They are characterized by the possibility of having points where the function is undefined (typically when the denominator equals zero) and often exhibit interesting properties such as vertical asymptotes and non-linear behavior across their domain.

Transcript

00:03 All right, so here's the function that we're given in the problem, and it represents the cost in dollars for removing x percent of pollutants from a bayou where there was a spill.

00:14 And so for part a, we're asked to find the cost for removing 20 percent of the pollutants.

00:21 So i'm calling that f of 20.

00:23 So we're going to substitute 20 in for x in the function, and then we're going to simplify it.

00:31 And that works out to be 6 million, oops, how about 2 million divided by 80, and that is $25 ,000.

00:50 Okay.

00:51 And in part b, we're asked to find the cost of removing 60 % and the cost of removing 80 % of the pollutants.

00:59 So we do the same thing.

01:00 For f of 60, we put 60 in for x, and we simplify.

01:11 And that gives us 6 million divided by 4%.

01:16 And that works out to be $150 ,000.

01:21 So $150 ,000.

01:23 For f of 80, same idea.

01:26 Substitute the 80 in there and simplify.

01:36 Here we have $8 million divided by 20.

01:40 And that works out to be $400 ,000.

01:46 Okay, and then we have part c.

01:48 And the idea of part c is to get us to think about what's happening as the percent of pollutants gets higher and higher and closer to 100 percent.

01:56 So we're finding f of 90 and f of 99.

02:07 All right, so we'll plug 90 into our function for x, 100 ,000 times 90, divided by 100 minus 90...

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