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Exponential and Logarithmic Functions Composite Functions Inverse 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 + 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 f(x) = 1/(x+2) g(x) = 4/(x-1) Find Find Find 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) = 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 f gof= 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. 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 f(x) = 3x + 4 g(x) = cube root of x 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 infinity as n approaches 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