Functions and InversesFunctions and InversesNumerical RepresentationNumerical Representation

Consider the ordered pairs: Consider the ordered pairs:

(-3, -5) (-2, -3) (-1,-1) (0,1) (1,3) (2, 5) (3, 7)(-3, -5) (-2, -3) (-1,-1) (0,1) (1,3) (2, 5) (3, 7)

The inverse is found numerically by switching The inverse is found numerically by switching the x and y coordinates:the x and y coordinates:

(-5, -3) (-3, -2) (-1,-1) (1,0) (3,1) (5, 2) (7, 3)(-5, -3) (-3, -2) (-1,-1) (1,0) (3,1) (5, 2) (7, 3)

Functions and InversesFunctions and InversesAnalytical RepresentationAnalytical Representation

The general rule/relationship of the ordered pairs is: The general rule/relationship of the ordered pairs is:

y = 2x + 1y = 2x + 1

The inverse is found analytically by switching x and y The inverse is found analytically by switching x and y in the equation and solving for y.in the equation and solving for y.

Test out some ordered pairs to see that this works!Test out some ordered pairs to see that this works!

This is just like finding the numerical inverse for This is just like finding the numerical inverse for all all pairspairs

1

2

xy

Functions and InversesFunctions and InversesGraphical RepresentationGraphical Representation

The graph of The graph of y = 2x + 1y = 2x + 1 is seen in blue. is seen in blue. The graph of The graph of is seen in red. is seen in red.

1

2

xy

A graph and its inverse are always symmetric about the line y =x

Example: Example: Graphical to NumericalGraphical to Numerical

State the inverse of the function shown as a set State the inverse of the function shown as a set of ordered pairs: of ordered pairs:

Graphical to NumericalGraphical to Numerical

We see that the original function has ordered pairs (-2,1)(-1,2)(2,-1) (3,1) and (4,4)

Thus, the ordered pairs for the inverse are (1,-2) (2,-1)(-1,2)(1,3) and(4,4)

Example:Example:Analytic (to Graphical) to AnalyticAnalytic (to Graphical) to Analytic

Find the rule for the inverse of the Find the rule for the inverse of the following function: following function:

y = y = x

Example:Example:Analytic (to Graphical) to AnalyticAnalytic (to Graphical) to Analytic

x y

Following the algorithm, students will likely say that the inverse is

Which corresponds to

2y xHowever, a quick sketch of the graph will show us why this is not entirely correct….

The (red) inverse is not a reflection of the function over the line

y = x

There is an extra branch!

We only want to use the part of the graph which is a reflection over the line y = x. This corresponds to the part of the red graph where the x coordinates are greater than or equal to zero.

x

2

0

y x

for x

The inverse of y=

Is actually

x

Without looking at the graph, most students will use both branches of the red graph, which is incorrect.

The previous conclusion could have also been reached The previous conclusion could have also been reached numerically:numerically:

Consider the ordered pair (-2, 4) which is a solution to Consider the ordered pair (-2, 4) which is a solution to the equation the equation 2y x

If (-2,4) makes the inverse equation true, then the point (4,-2) must make the original equation true,

2 4

This is not true because square roots are defined to be positive values. Therefore, the inverse we came up with must not be entirely correct!

Numeric to AnalyticNumeric to Analytic

State the inverse of the ordered pairs as an State the inverse of the ordered pairs as an equation where y depends on x.equation where y depends on x.

Assume there are infinitely many ordered pairs Assume there are infinitely many ordered pairs which follow this pattern:which follow this pattern:

(-2,-8) (-1,-1) (0,0) (1,1) (2,8) (3,27)(4,64)(-2,-8) (-1,-1) (0,0) (1,1) (2,8) (3,27)(4,64)

Numeric to AnalyticNumeric to Analytic

Students will need to Students will need to determine the rule for the determine the rule for the function which in this case function which in this case is is

Now, analytically the students Now, analytically the students will find the inverse to be will find the inverse to be

3y x

3y xA quick sketch of the graph will

show that no domain restrictions are necessary.

Numeric to AnalyticNumeric to Analytic

State the inverse of the ordered pairs as an State the inverse of the ordered pairs as an equation where y depends on x.equation where y depends on x.

Assume there are infinitely many ordered pairs Assume there are infinitely many ordered pairs which follow this pattern:which follow this pattern:

(5,-5) (3,-3) (2,-2) (0,0) (1,1) (2,2) (5,5)(5,-5) (3,-3) (2,-2) (0,0) (1,1) (2,2) (5,5)

Numeric to AnalyticNumeric to Analytic

Students often can not find a numerical Students often can not find a numerical representation for this relationship because it is representation for this relationship because it is not familiar. Encourage to represent the data not familiar. Encourage to represent the data graphically!graphically!

Numeric to AnalyticNumeric to Analytic

Many students will recognize this graph as having Many students will recognize this graph as having the “absolute value shape”. So the relationship the “absolute value shape”. So the relationship involve the absolute value function and can be involve the absolute value function and can be represented as represented as

The question asked for the inverse of the rule, which The question asked for the inverse of the rule, which is: is:

y = |x|y = |x|

y x

Choose the Best RepresentationChoose the Best Representation

Name a function with infinitely many ordered Name a function with infinitely many ordered pair solutions which is its own inverse. pair solutions which is its own inverse.

Stuck?Stuck?

This problem can easily be solved numerically This problem can easily be solved numerically because we are working with infinitely many because we are working with infinitely many ordered pairs.ordered pairs.

Stuck?Stuck?

This problem can easily be solved analytically This problem can easily be solved analytically because there are too many functions to be because there are too many functions to be able to easily guess and check.able to easily guess and check.

Think Graphically!Think Graphically!

All we need is a function whose graph is All we need is a function whose graph is symmetric about the line y = x!symmetric about the line y = x!

ExamplesExamples

y = xy = x

y = -xy = -x

y = 1/xy = 1/x

Use analytical, numerical, and graphical methods to verify that these equations are inverses of themselves!