In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.
This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable
x .
Consider the graph in
[link] of the function
Describe its overall shape. Is it periodic? How do you know?
The graph of
Using a graphing calculator or other graphing device, estimate the
- and
-values of the maximum point for the graph (the first such point where
x >0). It may be helpful to express the
-value as a multiple of π.
Now consider other graphs of the form
for various values of
A and
B . Sketch the graph when
A = 2 and
B = 1, and find the
- and
y -values for the maximum point. (Remember to express the
x -value as a multiple of π, if possible.) Has it moved?
Repeat for
A = 1,
B = 2. Is there any relationship to what you found in part (2)?
Complete the following table, adding a few choices of your own for
A and
B :
A
B
x
y
A
B
x
y
0
1
1
1
0
1
1
1
12
5
1
2
5
12
2
1
2
2
3
4
4
3
Try to figure out the formula for the
y -values.
The formula for the
-values is a little harder. The most helpful points from the table are
(
Hint : Consider inverse trigonometric functions.)
If you found formulas for parts (5) and (6), show that they work together. That is, substitute the
-value formula you found into
and simplify it to arrive at the
-value formula you found.
Key concepts
For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
For a function
and its inverse
for all
in the domain of
and
for all
in the domain of
Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
The graph of a function
and its inverse
are symmetric about the line
Key equations
Inverse functions
For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.
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