mrgilmartin.files.wordpress.com … · web viewinverse functions. we often know a function that...

Name: __________________________________ Block:__________ Date:________________ Paper:______

Inverse FunctionsWe often know a function that shows how to calculate values of one variable, y, from known

values of another variable, x. But problems in such situations often require discovering the value(s) of x that will produce specific target values of y.

Suppose you are given the following directions: From home, go north on Rt 23 for 5 miles Turn east (right) onto Orchard Street Go to the 3rd traffic light and turn north (left) onto Baldwin Avenue Jekoven’s house is the 5th house on the left. If you start from Jekoven’s house, write down the directions to get home.

How did you come up with the directions to get home from Jekoven’s?

Suppose you are given the following algorithm: Starting with a number, add 5 to it Divide the result by 3 Subtract 4 from that quantity Double your result Write a function f(x), which when given a number x (the original number) will model the operations given above.

Knowing the final result is 10, work backwards and find the original number. Show your work.

Write a function g(x), which when given a number x (the final result), will model the backward algorithm that you came up with above.

Fill out the following table: x ƒ(x) g(f(x)) ƒ(g(ƒ(x)))2217

-20What patterns do you notice in the columns (outputs)?

The plots of shown below. Notice the symmetry of the


two functions when plotted together.

Describe the symmetry in the graph above between the function and its inverse.

What is the equation for line of symmetry?

In this scenario, f(x) and g(x) are inverses of each other because g(x) will reverse the actions of f(x). Thus, we could write g(x) as f -1(y) described as “f inverse,” we typically write g=ƒ-1(x). The exponent “-1” does not mean the reciprocal “one over ƒ(x)” but “the inverse of ƒ(x).”

An inverse function takes and output and gives the corresponding input. Not all functions will have inverse functions for inverse functions must also pass the vertical line test

If the function ƒ uses x to calculate values of y, the ƒ-1 shows how to use values of y to calculate corresponding values of x. The challenge is using the rule for ƒ to find the rule for ƒ-1.

The daily profit of the Texas Theatre movie house is a function of the number of paying customers with rule P(n) = 5n-750

a. What numbers of paying customers will be required for the theater to have:i. A profit of $100?

ii. A profit of $1250?

iii. A loss of $175?

b. What rule can be used to calculate values for the inverse of the profit function – to find the number of paying customers required to reach any particular profit target?

Find the inverses of the following functions:a. If g(x)=3x then g(x)-1 = …

b. If k(x)=(3/5)x+4 then k(x)-1 = …

c. If s(x)=mx+b then s(x)-1 = …


What radical function f -1(y) would undo the actions of the function y=f(x)=x3+2 as shown in the table and plot below:

x y=f(x) f -1(y)

-1012

Solution: f -1(y)=___________Using the table: f -1( f (x))=___________

Line of symmetry: _______________

Study the following strategies for finding the inverse of to see if they are both correct.Strategy I Strategy 2

If Then

Then Swap roles of x and y

So,

If Swap the roles of y and x to get Then

Then

Hot air balloon flying is a popular recreation in several southwestern states. At some gatherings of balloonists, there is a competition to hit targets on the ground with bags dropped from balloons at various altitudes.

The balloons drift toward and over the target at various speeds. To have the best chance that their drop bags will hit the target balloonists need to know how fast the bags will fall toward the ground. Principles of physics predict that the velocity in feet per second will be a function of time in seconds with rule v(t)=32t. The distance fallen in feet will also be a function of time in seconds with rule d(t) = 16t2.

a. How long will it take the drop bag to reach a velocity of:i. 48 feet per second?

ii. 100 feet per second?

iii. 150 feet per second?


b. What rule can be used to calculate values for the inverse of the velocity function – to find the time when any given velocity is reached?

c. How long will it take the drop bag to reach the ground from altitudes of:i. 144 feet?

ii. 400 feet?

iii. 500 feet?

d. What rule can be used to calculate values of the inverse of the distance function – to find the time it takes for the drop bad to fall any given distance?

In some situations, you will need to find the inverse of a function that is defined only by table of values or a graph.a. Suppose that the function ƒ(x) is as shown in the following table. Make a similar table that shows

assignments bade by ƒ-1(x). Then describe the domain and range if ƒ(x) and ƒ-1(x).x -4 -3 -2 -1 0 1 2 3 4ƒ(x) 2 1 0 -1 -2 -3 -4 -5 -6ƒ-1(ƒ(x))

ƒ(x): Domain:___________ ƒ-1(x): Domain:___________ƒ(x): Range: ____________ ƒ-1(x): Range: ____________

b.

i. Plot points made by g-1(x). Then describe the domain and range of g(x) and g-1(x).g(x): Domain:___________ g-1(x): Domain:___________g(x): Range: ____________ g-1(x): Range: ____________

ii. Draw line segments connecting point (a, b) on the graph of g(x) to the corresponding point (b, a) on the graph of g-1(x). Then describe the pattern that seems to relate corresponding points on the graphs of g(x) and g-1(x).

Using what you have learned about the symmetry of ƒ(x) and f -1(x) graph f -1(x) for the following:


Why is y=x the line of symmetry for all of the above graphs?


Additional Practice

Using what you have learned about the symmetry of ƒ(x) and f -1(x) graph f -1(x) for the following:

Why is y=x the line of symmetry for all of the above graphs?

Adapt one strategy or the other to find rules for inverses of the following functions

a.

b.

c.

d.

mrgilmartin.files.wordpress.com … · web viewinverse functions. we often know a function that...

Documents