DAY 7 – INVERSE OF A FUNCTION

1. Use Exponential Regression to find an exponential function that contains the points (3, 54) and (4,162).

2. What is the initial value for this model?

3. What percentage growth or decay does this model imply?

WARM UP

KATHY AND KEVIN GRAPHED THE SAME DATA. BOTH INSIST THEY ARE CORRECT, BUT THEIR

GRAPHS LOOK DIFFERENT. WHAT DO YOU THINK HAPPENED?

LOOKING AT GRAPHS

Kathy and Kevin they switched their x and y values

In Kathy’s graph In Kevin’s Graph(0,1) (1,0) (2,4) (4,2)

WHAT HAPPENED?

In mathematics, the inverse of a function occurs when the independent and dependent values of a function are reversed.

We can create an inverse function by switching the x and y values.

(6, 2) will become (2, 6),

(-3, 1) becomes (1, -3).

When we find an inverse function, we have to make sure it is still a function.

INVERSE OF A FUNCTION

Remember, an inverse is an operation that take us back to the original input.

A function is a mathematical relation where each input only has one corresponding output.

Are each of these functions? Why or why not?

DO ALL FUNCTIONS HAVE INVERSES?

For a function to have an inverse, each output must only have one corresponding input.

Do these functions have inverses? Why or why not?

DO ALL FUNCTIONS HAVE INVERSES?

How f(x) = y and g(y) = x compare?

They have switched x and y.

Since x and y have switch places, we say that f and g are inverses.

A FUNCTION AND ITS INVERSE

If g is the inverse of f, we use the notation g = f -1 or g(x) = f -1(x).

The notation f -1 is read “f inverse of x.”

IMPORTANT NOTATION

1) f(x) = 6 + 3x 2) f(x) =  For each of the functions above, follow these stepsMake a table of 5 values and graph function 1 on graph

paper.Make a table of 5 values and graph function 2 the same

graph.What do you notice about the two tables?What do you notice about the two graphs?What line are the inverses refl ected over? Write your

conjecture on the line below.

To graph the inverse of a function, you can refl ect the original function over the line y = x OR make a table using the function, and then make a new table and switch the x and y coordinates. Now just graph the new table of points!

PART 1: GRAPHS OF INVERSE FUNCTIONS

)6(3

1x

1. Change f(x) notation to y notation1. So f(x) = 3 + x becomes y = 3 + x

2. Switch the x and the y variables in the function

3. Solve the equation for y.4. Replace y with

FINDING THE INVERSE EQUATION

)(1 xf

 You can check your work by putting your original and inverse functions in the calculator. If they are reflected over y = x, you’ll know you’ve done it right!

PART 2: EQUATIONS OF INVERSE FUNCTIONS

Complete the worksheet

HOMEWORK