APC
Unit 4
Composite functions
Inverse Functions

Warm-up
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Reflect on Your performance so far this year
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And on this latest Test
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What should you do differently?
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What should we do differently?
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How do we make class time more productive

5.1 Composite Functions
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(f o g)(x) = f(g(x))
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“f composed with g of x”
“f of g of x”
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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
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 f(g(1)) f(5) f(5) = 47
 g(f(1)) = -3 g(1) = 5

Algebraically
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Put the inner function into the outer function

Numerically

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

Recognizing Composition of functions
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Hint: Always look for the parenthesis

Question
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What is f(undefined)?
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Undefined so not included in the domain

To Determine the domain of
Composite Functions
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1. First, find the values to exclude from the inner function
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2. Compose the 2 functions
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Find the values excluded from the result
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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
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Verbally: one function undoes the other function.
Returns the value back to whatever you started with, x.
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Algebraically: “Show that 2 functions are inverses”
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(f o g)(x) =x
(g o f)(x) = x
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Numerically
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 f(x) = y f -1 (y)=x
(x,y)
(y,x)

You try it…

5.2 Inverse Functions
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Graphically
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Reflected over the line y=x
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To find the graph of the inverse Function
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Plot key points by switching the x and y values
Transform (x,y)  (y,x)

Worksheet #5

Worksheet #5 continued

Functions and Inverse
Functions
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Review: How do we know if a relation is a function
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Vertical line test
Each x has only one y
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Inverse Functions
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Reflected over y=x
The original function must pass the horizontal line test
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The inverse function (reflected) must pass the vertical line test
A function whose inverse is also a function is said to be:
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One-to-one
“Strictly Monotonic”
Increasing (or decreasing) over the entire domain

Worksheet #6

What’s rwong with this…?

How to find inverse Functions
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1. write using x and y notation
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Still the same function
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2. Switch the x and y variables
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This is no longer the same function
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(don’t use equals signs)
This is now the inverse function
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3. Solve for y
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Don’t forget to change back to inverse function notation

Try these

Help for the trickier problems
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Write using x,y
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Switch x and y
Move all the y terms to left side
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Move other terms to right side
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Factor out the y
Divide to solve for y

Start your Homework