Exponential and Logarithmic Functions Composite Functions Inverse Functions

  • Slides: 22
Download presentation
Exponential and Logarithmic Functions Composite Functions Inverse Functions Exponential Function Intro

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

Solving an Exponential Equation n If au = av, then u = v Get

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

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

Base e n E = (1 + 1/n)n infinity as n approaches n Look

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

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

Application Examples Page 426 #80, 88

Assignment n Page 397, 409, 423

Assignment n Page 397, 409, 423