APCUnit 4Composite functionsInverse Functions

Warm-up Reflect on Your performance so far this year

And on this latest Test

What should you do differently?

What should we do differently? How do we make class time more productive

5.1 Composite Functions (f o g)(x) = f(g(x))

“f composed with g of x” “f of g of x”

Evaluate or substitute from the inside out That is, evaluate the inner function, g(x)

first Then use that result to evaluate f(x)

Numerically

f(g(1)) g(1) = 5 f(5) f(5) = 47

g(f(1)) = -3

Algebraically

Put the inner function into the outer function

Numerically

Graphically:Find (g o f)(-1)Find (f o g)(2)

Recognizing Composition of functions Hint: Always look for the parenthesis

Question What is f(undefined)?

Undefined so not included in the domain

To Determine the domain of Composite Functions 1. First, find the values to exclude from

the inner function

2. Compose the 2 functions Find the values excluded from the result

The domain of the composite function is the set of values that excludes both.

Jump ahead to Problem #9 on worksheet

5.2 Inverse Functions Verbally: one function undoes the other function.

Returns the value back to whatever you started with, x.

Algebraically: “Show that 2 functions are inverses” (f o g)(x) =x (g o f)(x) = x

Numerically f(x) = y (x,y) f-1(y)=x (y,x)

You try it…

5.2 Inverse Functions Graphically

Reflected over the line y=x

To find the graph of the inverse Function Plot key points by switching the x and y

values Transform (x,y) (y,x)

Worksheet #5

Worksheet #5 continued

Functions and Inverse Functions Review: How do we know if a relation is a function

Vertical line test Each x has only one y

Inverse Functions Reflected over y=x The original function must pass the horizontal line test

The inverse function (reflected) must pass the vertical line test

A function whose inverse is also a function is said to be: One-to-one “Strictly Monotonic” Increasing (or decreasing) over the entire domain

Worksheet #6

What’s rwong with this…?

How to find inverse Functions 1. write using x and y notation

Still the same function

2. Switch the x and y variables This is no longer the same function

(don’t use equals signs) This is now the inverse function

3. Solve for y Don’t forget to change back to inverse function

notation

Try these

Help for the trickier problems Write using x,y Switch x and y Move all the y

terms to left side Move other terms

to right side Factor out the y Divide to solve for

y

Start your Homework