4-2:Composition and 4-2:Composition and Inverses of FunctionsInverses of Functions4-2:Composition and 4-2:Composition and Inverses of FunctionsInverses of Functions

English CasbarroEnglish CasbarroUnit 4Unit 4

f g x( ( )) f g x ( )

f g x( ( )) g f x( ( ))

Composition of Functions

The composition of functions is used when you have 2 functions. You will still do exactly the same thing as when you have a value, but you will now have another function. The notation for a composition is written 2 different ways:

or .

They are both stated: “ f of g , regardless of which notation you choose.

Note: and are 2 different things!

Recall that f(5) when f(x) = 2x – 4 is evaluated by substituting the value intothe equation. f(5) = 2(5) – 4 = 10 – 4 = 6, so f(5) = 6.

Be careful not to confuse the composition notation with the

notation for multiplication.fg(x) ≠ f(g(x))

Inverse Functions

If and are functions such that

then and are inverses of each other. The notation is given as such: If is the original function, then is the inverse of .

f g x g f x x( ( )) ( ( )

f

f gf

f 1f

If f and g are inverses of each other, then domain of f is the same as the range of g and vice versa.

Ex. Domain: {3,2, 5, –1} Range: {1,5,4,7}

The inverse of f:

f x( ) {( , ),( , ),( , ),( , )} 31 2 5 5 4 17

f x 1 13 5 2 4 5 7 1( ) {( , ),( , ),( , ),( , )}

Notice how the x and y have changed positions. Now the domain is {1,5,4,7} andthe range is {3,2,5,–1}. [This is also the reason that the horizontal line test works]

When you are finding the inverse of a function, you will still switch the x and the y, but then you will solve for the y. For example, find the inverse of f(x)=3x + 2

y x 3 2

x y 3 2

Step 1: Write out the function using a y instead of function notation:

Step 2: Switch the x and the y:

Step 3: Solve for y:x y 3 2 2 2x y 2 3

(divide everything by 3)

yx

23

f xx

1 2

3( ) f x

x 1

3

2

3( )or or

When the function is not linear, you will still follow the procedure

Find the inverse of f x x( ) 2 52

If the function is exponential, you will need to define another termEx. y = 2x In order to solve the inverse equation for y,

you will need a logarithm.

The inverse of y = 2x is log2x = y

Logarithms

Definition: A logarithm is an exponent to which a specific base has beenraised to obtain a specific value.

Exponential form Logarithmic form

You can write an exponential equation in logarithmic form and vice versa.

Exponential Form

Logarithmic Form

26=64 log264=6

41=4 log44=1

5-2=0.04 Log50.04=-2

50=1 log51=0

34=81 log381=4

Converting between exponential and logarithmic form

Solving by using mental math

log2x=4

log39=x

logx27=3

log168=¾

Turn in the following Problems1. A theater sells tickets for $22. If you pay by credit card, the theater adds a service charge of $3.50 to the entire order. a. Write a function that gives the amount billed to the credit card as a function of the number of tickets purchased. b. Write the inverse function, and use it to find the number of tickers purchased when the credit card bill is $157.50 c. Is it possible to have a total of $332.50 billed to your credit card for these tickets? Why or why not?

2. For a certain credit card with 19% annual interest compounded monthly, the total amount A that you owe after n months is given by A = P(1.016)n , where P is the starting balance.

a. You start with a balance of $500. Write and solve a logarithmic expression for the number of months it will take for the debt to double. b. How many additional months will it take for the debt to double again? c. Does the amount of time that it takes the debt to double depend on the starting balance?