Exponential and Logarithmic Functions Composite Functions Inverse Functions
- Slides: 22
Exponential and Logarithmic Functions Composite Functions Inverse Functions Exponential Function Intro
Objectives n n n 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 n n n 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. n 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) = 2 x and g(x) = 3 x 2 + 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) n Find the domain of the composite
f(x) = 1/(x+2) g(x) = 4/(x-1) n n n Find Find the domain of the composite f o g fog the domain of the composite g o f g of (g o f)(4) Find the domain of f o g if f(x) = square root of x and g(x) = 2 x + 3
Find the components of the following composites: n H(x) = (x 2 + 1)50 n S(x) = 1 / (x + 1)
Show that the two composite functions are equal for: n f(x) = 3 x – 4 n f n gof= n Look at number 8 on page 397 o g(x) = (1/3)(x + 4) g=
When both composites end up with x as the final range they are inverse functions. n n 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 n f(x) = x 3 n f(x) = 3 x + 4 g(x) = cube root of x f-1(x) = (1/3)(x – 4)
Finding inverses n n n 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 n n 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 n as. at = as+t n (as)t = ast n (ab)s = as. bs n 1 s = 1 n a 0 = 1 n a-s = 1/as
Graphs of Exponential Functions n f(x) = (1/2)x f(x) = 2 x n Plug numbers in for x and graph n Look at function values at f(1) n Look at bases: what happens when base is fraction? When base is whole value? n As base gets bigger – what happens to graph?
Transformations: work same as on quadratic n F(x) = 3 -x + 2 n n n Up 2, reflect across x-axis Horizontal asymptote at y=2 F(x) = 2 x-3 – 5 Right 3, down 5 Horizontal asymptote at y=-5
Examples n Page 423, #15, 23, 31, 34, 44, 74
Solving an Exponential Equation n If au = av, then u = v Get bases equal, then set exponents equal and solve. 3 x+1 = 81
More examples n Page 425; #54, 58, 62, 68, 66
Base e n E = (1 + 1/n)n infinity as n approaches n Look at Page 419 – bottom of page n Approximate value? n Called the natural base
Graph: n F(x) = ex n F(x) = -ex-3 n Look at translations n n n 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 n Page 397, 409, 423
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