1. 4-2:Composition and Inverses of Functions English Casbarro Unit 4

2. 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 into the equation. f(5) = 2(5) – 4 = 10 – 4 = 6, so f(5) = 6.

5. Be careful not to confuse the composition notation with the notation for multiplication. fg(x) ≠ f(g(x))

9. 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.

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

12. 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 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: (divide everything by 3) or

14. When the function is not linear, you will still follow the procedure  Find the inverse of

15. If the function is exponential, you will need to define another term Ex. y = 2 x In order to solve the inverse equation for y, you will need a logarithm. The inverse of y = 2 x is log 2 x = y

16. Logarithms Definition: A logarithm is an exponent to which a specific base has been raised to obtain a specific value. Exponential form Logarithmic form You can write an exponential equation in logarithmic form and vice versa.

17. Exponential Form Logarithmic Form 2 6 =64log 2 64=6 4 1 =4log 4 4=1 5 -2 =0.04Log 5 0.04=-2 5 0 =1log 5 1=0 3 4 =81log 3 81=4 Converting between exponential and logarithmic form

18. Solving by using mental math  log 2 x=4  log 3 9=x  log x 27=3  log 16 8=¾

19. Turn in the following Problems 1.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?