Exponential and Logarithmic Functions

Composite FunctionsInverse Functions

Exponential Function Intro

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’

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.

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.

Examples: Suppose f(x) = 2x and g(x) = 3x2 + 1Find (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

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

Find the components of the following composites:

H(x) = (x2 + 1)50

S(x) = 1 / (x + 1)

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

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

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)

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

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

Laws of Exponents as . at = as+t

(as)t = ast

(ab)s = as . bs

1s = 1

a0 = 1

a-s = 1/as

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?

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

Examples

Page 423, #15, 23, 31, 34, 44, 74

Solving an Exponential Equation

If au = av, then u = v

Get bases equal, then set exponents equal and solve.

3x+1 = 81

More examples

Page 425; #54, 58, 62, 68, 66

Base e

E = (1 + 1/n)n as n approaches infinity

Look at Page 419 – bottom of page

Approximate value?

Called the natural base

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

Application Examples

Page 426 #80, 88

Assignment

Page 397, 409, 423