1. Exponential and Logarithmic Functions
Composite Functions
Inverse Functions
Exponential Function Intro

2. Objectives Form a composite function and find its domain
Determine the inverse of a function
Obtain the graph of the inverse from the graph of a function
Evaluate and graph an exponential function
Solve exponential equations
Define the number ‘e’

3. Composite Functions
Combining of two or more processes into one function
(f o g)(x) = (f(g(x))) = read as “f composed with g”
The domain is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

4. Look at diagrams on page 392 of text book.
In figure 1, the top value of x would not be in the composite domain since the range of g does not exist in the domain of f.

5. Examples: Suppose f(x) = 2x and g(x) = 3x2 + 1 Find (f o g)(4)
Find (g o f)(2)
Find (f o f)(1)
Find (f o g)(x)
Find (g o f)(x)
Find the domain of the composite

6. f(x) = 1/(x+2) g(x) = 4/(x-1)
Find the domain of the composite f o g
Find f o g
Find the domain of the composite g o f
Find g o f
Find (g o f)(4)
Find the domain of f o g if f(x) = square root of x and g(x) = 2x + 3

7. Find the components of the following composites:
H(x) = (x2 + 1)50
S(x) = 1 / (x + 1)

8. Show that the two composite functions are equal for:
f(x) = 3x – 4 g(x) = (1/3)(x + 4)
f o g =
g o f =
Look at number 8 on page 397

9. To Test: Each of the following must be true
When both composites end up with x as the final range they are inverse functions.
Inverse functions: when a function manipulates the range of one function and outputs the original domain
To Test: Each of the following must be true
(f o g)(x) = x
(g o f)(x) = x

10. Determine if the following functions are inverses
f(x) = x3 g(x) = cube root of x
f(x) = 3x + 4 f-1(x) = (1/3)(x – 4)

11. Finding inverses Ordered Pairs: reverse the x and y
Equations: reverse x and y then solve for y
Graphs: Invert x’s and y’s off of original graph, plot new points

12. Exponential Functions
f(x) = ax
a is a positive real number a ≠ 0, domain is the set of all real numbers
a: is called the base number
x: is called the exponent

13. Laws of Exponents as . at = as+t (as)t = ast (ab)s = as . bs 1s = 1
a-s = 1/as

14. Graphs of Exponential Functions
f(x) = (1/2)x f(x) = 2x
Plug numbers in for x and graph
Look at function values at f(1)
Look at bases: what happens when base is fraction? When base is whole value?
As base gets bigger – what happens to graph?

15. Transformations: work same as on quadratic
F(x) = 3-x + 2
Up 2, reflect across x-axis
Horizontal asymptote at y=2
F(x) = 2x-3 – 5
Right 3, down 5
Horizontal asymptote at y=-5

16. Examples
Page 423, #15, 23, 31, 34, 44, 74

17. Solving an Exponential Equation
If au = av, then u = v
Get bases equal, then set exponents equal and solve.
3x+1 = 81

18. More examples
Page 425; #54, 58, 62, 68, 66

19. Base e E = (1 + 1/n)n as n approaches infinity
Look at Page 419 – bottom of page
Approximate value?
Called the natural base

20. Graph: F(x) = ex F(x) = -ex-3 Look at translations
Same as translations for other functions
Add/Subtract after base: vertical shift
Add/Subtract in process: horizontal shift
Negative: reflection
Numbers multiplied: Stretch/Compression

21. Application Examples
Page 426 #80, 88

22. Assignment
Page 397, 409, 423