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