1. AP CALCULUS AB Chapter 1: Prerequisites for Calculus Section 1.5: Functions and Logarithms

2.  One-to-One Functions  Inverses  Finding Inverses  Logarithmic Functions  Properties of Logarithms  Applications …and why Logarithmic functions are used in many applications including finding time in investment problems. What you’ll learn about…

3. One-to-One Functions  A function is a rule that assigns a single value in its range to each point in its domain.  Some functions assign the same output to more than one input.  Other functions never output a given value more than once.  If each output value of a function is associated with exactly one input value, the function is one- to-one.

4. One-to-One Functions

6. Inverses  Since each output of a one-to-one function comes from just one input, a one-to-one function can be reversed to send outputs back to the inputs from which they came.  The function defined by reversing a one-to-one function f is the inverse of f.  Composing a function with its inverse in either order sends each output back to the input from which it came.

7. Slide 1- 7 Inverses

8. Identity Function The result of composing a function and its inverse in either order is the identity function.

9. Example Inverses

10. Writing f -1 as a Function of x.

11. Section 1.5 – Functions and Logarithms  A function is an inverse of another function if and only if for all x in the domain.  To find the inverse of a one-to one function: 1. Interchange x and y. 2. Solve for y.

12. Finding Inverses

13. Example Finding Inverses [-10,10] by [-15, 8]

14. Section 1.5 – Functions and Logarithms  Ex:

15. Section 1.5 – Functions and Logarithms  You try: Show that the function is one-to-one and find its inverse function.

16. Section 1.5 – Functions and Logarithms  To graph the inverse parametrically: If is a function that can be defined parametrically as then the inverse can be defined as

17. Base a Logarithmic Function

18. Logarithmic Functions  Logarithms with base e and base 10 are so important in applications that calculators have special keys for them.  They also have their own special notations and names.

19. Inverse Properties for a x and log a x

20. Properties of Logarithms

21. Example Properties of Logarithms

23. Slide 1- 23 Change of Base Formula

24. Slide 1- 24 Example Population Growth

25. Section 1.5 – Functions and Logarithms  You try: Solve for x: 1. 2.

26. Section 1.5 – Functions and Logarithms  You try: Sulmaz invests $18000 in an account that earns 6.75% interest compounded annually. How long will it take the account to reach $4320?