Day 7 – Inverse of a Function

Warm Up What is the exponential function that contains the points (3, 54) and (4,162). What is the initial value for this model? What percentage growth or decay does this model imply?

How are functions and their inverses related? Essential Question #6? How are functions and their inverses related?

Looking at Graphs Kathy and Kevin graphed the same data. Both insist they are correct, but their graphs look different. What do you think happened? Ask students to list and think about the points

Kathy and Kevin they switched their x and y values What happened? 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)

Inverse of a Function 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.

Do all functions have inverses? 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? Discuss first, then the definition will fly in.

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? Does f(x) = x have an inverse? In the mapping, does an input of -3 always give an output of 3? Does an output of 3 always come from an input of -3? How does the diagram show reasons for your answer? Does g(x) = x + 3 have an inverse? How does the diagram show reasons for your answer? Let’s see, we have the ordered pairs (1,3), (2,4), and (3,6). The inverse of those ordered pairs are (3,1), (4,2), and (6,3). All you have to do is just reverse the direction of the arrow, so g(x) does have an inverse.

A function and its inverse 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.

IMPORTANT NOTATION 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.”

Part 1: Graphs of Inverse Functions 1) f(x) = 6 + 3x 2) f(x) =   For each of the functions above, follow these steps Make 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 reflected over? Write your conjecture on the line below. To graph the inverse of a function, you can reflect 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!

Finding the inverse Equation Change f(x) notation to y notation Switch the x and the y variables in the function Solve the equation for y. Replace y with

Part 2: Equations of Inverse Functions 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!

Homework Complete the worksheet