Inverses

Table of ContentsFinding the inverse …..…… Slides 2 – 16Finding (f of f-1)(x) ……….. Slides 17 – 22Is it function? ………………. Slides 23 – 25Quick Assessment …..………..….. Slide 26

Answer Key …………………………… Slide 27

What is an Inverse?When we say the Inverse we are talking about the Inverse relation.

An Inverse relation is where the X and Y coordinate values are reversed.

Given (2, -6) ~ the inverse is (-6, 2)Given y = x ~ the inverse is x = y

Example 1:Find the Inverse of the relation

Relation Inverse Relation

-5 2

4 3

2 2

-1 -5 9 6

X Y

2 -5

3 4

2 2

-5 -1 6 9

x y

To find the inverse of a set of points:

Simply….

switch x’s and y’s

x -7 -1 0 4 32 163

y -12 -8 -4 -1 8 63

Example 2: Find the Inverse of the relation given a table

Relation

Inverse Relation

x -12 -8 -4 -1 8 63

y -7 -1 0 4 32 163

x -2 0 4 8

y -5 -5 2 7

When given a set of points in a table, you can determine if it representsa function by looking at the x values.

If the x values repeat, then the table only represents a relation. If the x values do not repeat, then it represents a function.

The table above represents a function.

Below is its inverse,

x -5 -5 2 7

y -2 0 4 8

The inverse is a relation, because its x valuesdo repeat.

Relation: ( -2, 5), (0, 3), (2, 0), (4, -2)

Inverse: ( 5, -2), (3, 0), (0, 2), (-2, 4)

When we find the inverse, interchange the x and y values of each ordered pair, the domain of the relation becomes the range of the inverse, and the range of the relation becomes the domain of the inverse.

Relation: ( -2, 5), (0, 3), (2, 0), (4, -2)Domain: (-2, 0, 2, 4)Range: (-2, 0 , 3, 5)

Inverse: ( 5, -2), (3, 0), (0, 2), (-2, 4)Domain: (-2, 0 , 3, 5)Range: (-2, 0, 2, 4)

On a piece of graph paper, graph the relation and the inverse.What do you notice about the two graphs?

Example 3: Find the Inverse of the relation given a set of ordered pairs

Original Inverse

What if it’s an equation?We’ve looked at finding inverses when you have a tableor a given relation, but what if you have an equation?

You use the same concept, but now you’re going to work with the variables. First, swap the x and the y variables. Then re-solve for y. That’s it!

Let’s look at an example!

Example 4:Find the inverse of f(x) = 2x + 5 and then find the intersection point of the f-1 (x) and f(x).

Swap x and y (remember f(x) is another way to say y!):x = 2y + 5

Resolve for y: x = 2y + 5 -5 -5

x – 5 = 2y 2 2

x – 5 = y 2

So f-1 (x) = (x – 5) / 2

Notice how all the operations switched to their inverse operations!?

This the symbol for inverse!

Therefore, to find the inverse of an equation:~Make f(x) a y~Switch x and y~Solve for y

Now for the intersection point…To find the intersection point you have two options:1. Graph both of the equation, or2. Set the equation equal to each other and solve for x.

Graph: Algebra:Set equal to each other

2x + 5 = (x-5)/2Multiply every term by 2 to eliminate fraction

4x + 10 = x – 5Solve for x

3x = -15 x = -5

Plug in x to find y 2(-5) + 5 = y

-10 + 5 = y -5 = y

Recall f(x) = 3x – 6 y = 3x - 6.

 Interchange the x and the y.

x = 3y – 6 

Solve for y.(We want to be able to rewrite the inverse in function notation!)

 x + 6 = 3y

Divide both sides by 3

yx

36

Hence, the inverse is = )(1 xf

36x

Another Example….

ActivityGo to http://www.calculateforfree.com/graph.html(Go down to Graphing Calculator Click – “Click to Start Gcalc” button –(Type equations in the top text box)(You can click reset to enter a new set of equations)

Enter the follow functions: f(x) = x + 3 and g(x) = x – 3

What do you notice about these graphs? (Hint what is their relation around the line y = x)

Try the following relation and its inverse relation:f(x) = 2x and g(x) = 1/2x

What do you notice about the ordered pairs for each set of graphs? What do you notice about the graph of the lines f(x) and g(x)? (Consider symmetry in your response.) What do you notice about the equations of f(x) and g(x)?

What do you notice about the ordered pairs for each set of graphs?

What do you notice about the graph of the lines f(x) and g(x)? (Consider symmetry in your response.)

What do you notice about the equations of f(x) and g(x)?

The inverse relations ordered pairs have the x and y values switched.

The graphs of f(x) and g(x) are symmetrical about the line y = x.You could also say the graphs are mirror images of each other about the line y = x.

f(x) and g(x) are inverses of each other.

Answers

On the following slides are many example of how to find the inverse given the equation!

Some are easy and some are a little more difficult…

Check them out!

Given: f(x) = 3x + 8

Find f-1(x): (f(x) inverse, or, the inverse of f(x))

To find the inverse of a function:

1)Change f(x) to y

2)Make all x’s be y, and all y’s be x

3)Solve for y

4)Replace y with f^-1(x)

f(x) = 3x + 8y = 3x + 8

x = 3y + 8

x – 8 = 3y

x – 8 x - 8x - 8------- = y => f^-1(x) = -----------f^-1(x) = ----------- 3 33

Given: f(x) = x3 - 4

Find f-1(x): (f(x) inverse, or, the inverse of f(x))

To find the inverse of a function:

1)Change f(x) to y

2)Make all x’s be y, and all y’s be x

3)Solve for y

4)Replace y with f^-1(x)

f(x) = x3 - 4y = x3 - 4

x = y3 - 4

x + 4 = y3

= y3 4 x

31 4)( xxf

Given: f(x) = 5x + 11

Find (f o f-1)(x):

(f o f^-1)(x) means: Put the expression for f^-1(x) into all x’s in f(x)

1)Find the inverse: f^1(x) = (x -11) / 52)Plug the inverse into all x’s in f(x):

Since, f(x) is 5x + 11 and f^-1(x) is (x-11) / 5

xxfof

xxfof

xxfof

xxfof

xxfof

simplified

xxfof

))((

5

5))((

5

55555))((

5

55

5

555))((

115

555))((

.....

115

115))((

1

1

1

1

1

1

Given: f(x) = x - 9

Find (f o f^-1)(x):

(f o f^-1)(x) means: Put the expression for f^-1(x) into all x’s in f(x)

1)Find the inverse: f^1(x) = x + 92)Plug the inverse into all x’s in f(x):

Since, f(x) is x - 9 and f^-1(x) is x + 9

xxfof

simplified

xxfof

))((

.....

9)9())((

1

1

Given: f(x) = x - 9

Find (f o f^-1)(-8):

1)Find the inverse: f^1(x) = x + 92)Plug the inverse into all x’s in f(x):

3)Plug -8 into the x’s in the final answer

From above…

8)8)((

)8)((

.....

9)9()8)((

1

1

1

fof

xfof

simplified

xfof

The relationship between Composites and Inverse Relations

231

)(

63)(

1

xxf

xxfFind and ))(())(( 11 xffxff

))(( 1 xff 6)231(3 x = x + 6 – 6 = x

))((1 xff 2)63(3

1x x – 2 + 2 = x

If f and g are functions and , then f and g areInverses of each other.

xxfgxgf ))(())((

Determine whether the following functions are inverses of one another by

finding ))(())(( xfgandxgf

451

)(

205)(

3)(

3)(

5)(

5)(

25)(

32)(

xxg

xxf

xxg

xxf

xxg

xxf

xxg

xxfNo

Yes

Yes

Yes

Every year the NBA holds All-Star game. A limited amount of players are asked to participate in the slam dunk contest based on their performance throughout the season.  A basketball hoop is 10 feet above the floor. A player must jump high enough to reach 10 ½ feet to make a slam dunk. A person’s height J in feet can be modeled by where R is the person’s standing reach in feet.

13

2 RH

1. Write the inverse of the function. 2. Find the standing reach of a person with a height of 6 feet.3. Find the standing reach of a person with a height of 6 feet and 6 inches.

1.

2. 7 feet 6 inches

3. 8 feet 3 inches

HR

2

33

Recall, the graphs of inverses are “mirror images” or reflections of one another across the line y = x.

This means every point (x,y) on the graph f(x) corresponds to a point (y,x) on the graph .

This is true for any two relations that are inverses of each other.

)(1 xf

Before, we used the Vertical Line Test to see if a relation was a function.

Now we use the Horizontal Line Test to see if the inverse is a function.

The horizontal line test is done by “drawing” a horizontal line through a graph. If the line passes through 2 or more points on the graph, then the graph’s inverse is not a function.

If the graph does pass the horizontal line test, Then the function is one-to-one and the Inverse is a function.

http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/geometry/vertandhorizlinetests.html

http://www.mathwarehouse.com/algebra/relation/one-to-one-function.php

Quick Assessment1. Find the inverse of :

2. List the Domain and Range of the inverse of:

3. Find f-1(x) if f(x) = 3x2 - 2.4. Find (f of f-1)(12).5. Are y = 4x – 3 and y = 4x + 3

inverses?

x 2 5 8

y -1 5 12

x 3 15 18

y -11 -21 -25

Answer Key1. Inverse is

2. Domain: -11, -21, -25 Range: 3, 15, 183. Find f-1(x) if f -1(x) = .4. (f of f-1)(12) = 12.5. NO, because 4x – 3 = 4x + 3 gives you

-3 = +3 which is a false statement! The correct inverse of 4x – 3 would be y = (x + 3) / 4

x -1 5 12

y 2 5 8

x3

2