Inverses of Functions Section 1 6 Objective To

Inverses of Functions Section 1. 6

Objective To be able to find inverses of functions. n

Relevance To be able to model a set of raw data after a function to best represent that data. n

Go Over Quiz

Warm Up: Graph & give the domain & range. Answer on Next Slide

Warm Up #3: Graph & give the domain & range. x y -3 -4 -5 2 1 0 x y 1 -3 2 -2 3 -1 4 0

Intro to Inverses n n n Sneetches Video or Read Story Quickly A function describes the relationship between 2 variables, applying a rule to an input that generates exactly one output. For such relationships, we are often compelled to “reverse” or “undo” the rule.

The Sneetches Story n n Some of the Sneetches have stars on their yellow bellies; some do not. In the story, Sylvester Mc. Monkey Mc. Bean builds a machine that can apply these stars. Sneetches without stars pay Mc. Bean to make them look like their start-bellied friends. When all Sneetches appear the same, another problem emerges. So………………

The Sneetches Story n n Mc. Bean builds a second machine to “reverse” the action of the first one – one that removes the stars. Can you identify the math connections to this story?

Math Connections n n functions Mc. Bean’s machine represents______. The second machine “reverses” the action of inverse the first, thereby acting as an ______.

Progress of Inverses Throughout Math n n Learned Addition and then its inverse operation Subtraction. Learned Multiplication and then its inverse operation Division. Learning Perfect Squares connects with extracting Square Roots Basically – Inverses are a second operation that reverses the first one!

Calculator Inverses n n n Take a look at your GDC’s and observe the keys. Do you notice that inverse operations of many calculator commands are “second” functions? A calculator key pairs an operation or function with its inverse.

Inverse Function Partner Share. List the sequence of steps needed to evaluate this function: 1. Square the Input 2. Multiply by 9 3. Add 4

Inverse Functions Partner Share Divide into groups of 3 or 4. n Each group will be given two functions. n For each function: 1) List the steps of each function 2) Provide 3 ordered pairs that will satisfy the function. n Seek out other teams who had the other half of the function share information. For example, a team that had 2 A will seek out a team that had 2 B. n Class Discussion – What do you notice about the “undoing” of inverses of functions? n

Warm Up #3: Graph & give the domain & range. Answer on Next Slide

Warm Up #3: Graph & give the domain & range. x y -3 -4 -5 2 1 0 x y 1 -3 2 -2 3 -1 4 0

Inverse of a relation n The inverse of the ordered pairs (x, y) is the set of all ordered pairs (y, x). The Domain of the function is the range of the inverse and the Range of the function is the Domain of the inverse. Symbol: In other words, switch the x’s and y’s!

Example: {(1, 2), (2, 4), (3, 6), (4, 8)} Inverse:

Function notation? What is really happening when you find the inverse? Find the inverse of f(x)=4 x-2 x *4 -2 (x+2)/4 /4 +2 So 4 x-2 x

To find an inverse… n Switch the x’s and y’s. n Solve for y. n Change to functional notation.

Find Inverse:

Draw the inverse. Compare to the line y = x. What do you notice?

Graph the inverse of the following: x y 0 -5 -3 -4 1 2 1 4 The function and its inverse are symmetric with respect to the Line y = x.

How Does an Inverse Exist n

More on One-To-One n n Recall that a function is a set of ordered pairs where every first coordinate has exactly one second coordinate. A one-to-one function has the added constraint that each 2 nd coordinate has exactly one 1 st coordinate.

Example n

Things to note. . Ø Ø The domain of is the range of f(x). The graph of an inverse function can be found by reflecting a function in the line y=x. Check this by plotting y = 3 x + 1 and on your graphic calculator. Take a look

Reflecting. .

Find the inverse of the function. Is the inverse also a function? Let’s look at the graphs. Inverse NOTE: Inverse is NOT a function!

Horizontal Line Test n n Recall that a function passes the vertical line test. The graph of a one-to-one function will pass the horizontal line test. (A horizontal line passes through the function in only one place at a time. )

Is it an Inverse? A function can only have an inverse if it is one-to-one. You can use the horizontal line test on graphical representations to see if the function is one-to-one.

Composition and Inverses n If f and g are functions and then f and g are inverses of one another. !!!!!!!!!!!

Example: Show that the following are inverses of each other. The composition of each both produce a value of x; Therefore, they are inverses of each other.

Are f & g inverses?

You Try…. n Show that n are inverses of each other.

Are f & g inverses?