Section 6.1: Composite Functions
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1 Section 6.1: Composite Functions Def: Given two function f and g, the composite function, which we denote by f g and read as f composed with g, is defined by (f g)(x) = f(g(x)). In other words, the function f composed with g is the function you get by putting the function g into the function f. The domain of f g is the set of all x in the domain of g such that g(x) is in the domain of f. ex. Given f(x) = x + 1 and g(x) = 3x, find: a) (f g)(8) b) (g f)(3) c) (f f)(15) d) (g g)(1) ex. For the given function, find: a)f g, b)g f, c)f f, d)g g, and state the domain of each: 1
2 i) f(x) = 1 2, g(x) = x+3 x ii) f(x) = x 2 + 4, g(x) = x 2 ex. Find functions f and g such that f g = H. a) H(x) = (3x 2 + 2x 1) 4 b) H(x) = 4x 2 9 c) H(x) = 3x 5 2
3 Section 6.2: One-to-One Functions; Inverse Functions Def: Suppose that f is a function. The inverse of f is the correspondence which takes f(x) as the input and gives back x as the output. The domain of f is the range of the inverse of f and the range of f is the domain of the inverse of f. ex. Find the inverse of the following functions and determine whether the inverse is a function. a) b) c) {(1, 2), (2, 8), (3, 18), (4, 32)} Def: When the inverse of a function is itself a function, then we call f a oneto-one function. In other words, f is a one-to-one function if, for any choice of elements x 1 and x 2 in the domain of f, with x 1 x 2, the corresponding values f(x 1 ) and f(x 2 ) are not equal in the range of f. Horizontal-line Test: If every possible horizontal line that you can draw in the xy-plane intersects the graph of a function f in at most one point, then f is one-to-one. ex. Determine whether the following functions are one-to-one or not: 1
4 a) f(x) = (x 1) 2 b) g(x) = x
5 c) Notation: The inverse of f is denoted by f 1. Note that this does not mean 1 f(x). Facts: i) Domain of f = Range of f 1 and Range of f = Domain of f 1 (we stated this in the definition of the inverse of f, but now we can write it using our notation of f 1 ). ii) f(f 1 (x)) = x and f 1 (f(x)) = x ex. Verify that the functions f and g are inverses of each other. a) f(x) = 3x + 4; g(x) = 1 (x 4) 3 b) f(x) = (x + 3) 2, x 3; g(x) = x 3 3
6 c) f(x) = 2 x 2 3x ; g(x) = 3+x 1+x Theorem: The graph of a function f and the graph of its inverse f 1 are symmetric with respect to the line y = x. Symmetry about the line y = x means the x- and y-coordinates are switched. ex. The graph of a one-to-one function f is given. Draw the graph of f 1. The graph of y = x is also already given. a) 4
7 b) To find the inverse of the function y = f(x), interchange the x and y to get x = f(y). Then, if possible, solve for y. This will give you f 1. ex. The function f is one-to-one. Find its inverse. State the domain and range of f and f 1. a) f(x) = x b) f(x) = 3x x+1 5
8 c) f(x) = 3x+2 x 6 d) f(x) = x2 +3 3x 2, x > 0 6
9 Section 6.2: Example Answers ex. Determine whether the following functions are one-to-one or not: a) f(x) = (x 1) 2 b) g(x) = x
10 c) ex. The graph of a one-to-one function f is given. Draw the graph of f 1. The graph of y = x is also already given. a) 2
11 b) 3
12 Section 6.3: Exponential Functions Def: An exponential function is a function of the form f(x) = a x where a is a positive real number and a 1. The domain of f is the set of all real numbers. Exponent Laws: a m a n = a m+n (a m ) n = a mn (ab) n = a n b n a m/n = n a m = ( n a) m ( ) a n b = a n b n a n = 1 a n 1 a n a 0 = 1 = a n 1 n = 1 Properties of the Exponential Function f(x) = a x, a > 1: 1. The domain is the set of all real numbers. The range is the set of positive real numbers. 2. There are no x-intercepts and the y-intercept is The line y = 0 (the x-axis) is the horizontal asymptote as x. 4. f is always increasing and is therefore one-to-one on (, ). 5. The graph of f contains the points (0, 1), (1, a), ( 1, 1 a ). 6. Graphs of the exponential function f(x) = a x for a = 2, 3, 4. 1
13 Properties of the Exponential Function f(x) = a x, 0 < a < 1: 1. The domain is the set of all real numbers. The range is the set of positive real numbers. 2. There are no x-intercepts and the y-intercept is The line y = 0 (the x-axis) is the horizontal asymptote as x. 4. f is always decreasing and is therefore one-to-one on (, ). 5. The graph of f contains the points (0, 1), (1, a), ( 1, 1 a ). 6. Graphs of the exponential function f(x) = a x for a = 1 2, 1 3, 1 4. Just like we use π to symbolize the number π , use use e to symbolize the number e The number e, just like π, is very important in mathematics and comes up often in applications. We call the function f(x) = e x the exponential function, even though any function of the form f(x) = a x, where a is any positive real number, is an exponential function. ex. Graph the function then state the domain, range, and horizontal asymptote: 2
14 a) f(x) = e x 1 b) f(x) = 3e x+2 3
15 c) f(x) = 9 2 x Fact: If a u = a v, then u = v. ex. Solve: a) 5 1 2x = 1 5 b) 8 x2 2x = 1 2 4
16 c) 4 x2 = 2 x d) ( 1 2) x = 4 e) 4 x 2 x = 0 ex. If 2 x = 3, what does 4 x equal? 5
17 Section 6.3: Example Answers ex. Graph the function then state the domain, range, and horizontal asymptote: a) f(x) = e x 1 b) f(x) = 3e x+2 1
18 c) f(x) = 9 2 x 2
19 Section 6.4: Logarithmic Functions Def: The logarithmic function to the base a > 0, denoted by y = log a x and read as log base a of x, is the inverse function of the exponential function y = a x. log a x is defined to be the exponent that a needs to have in order to give you the value x. In other words, y = log a x is equivalent to writing x = a y. The domain of y = log a x is x > 0 and the range is (, infty). Notation: We denote log e by ln and call it the natural logarithm. We denote log 10 by log (so if you just see log without any base specified, then it automatically means base 10) and call it the common logarithm. ex. Change the exponential expression to an equivalent expression involving a logarithm: a) 64 = 8 2 b) 2 x = 6.2 c) = N ex. Change the logarithmic expression into an equivalent expression involving an exponent: a) log 4 ( 1 64) = 3 b) log 3 8 = x c) ln x = 2.1 ex. Find the exact value: 1
20 a) log 6 6 b) log c) log 2 4 d) ln e 4 The logarithmic function is the inverse of the exponential function, so the domain of the logarithmic function is the same as the range of the exponential function, which is (0, ), and the range of the logarithmic function is the same as the domain of the exponential function, which is (, ). ex. Find the domain: a) g(x) = ln(x 4) ( b) h(x) = log x ) 3 x 3 2
21 Since the logarithmic function is the inverse of the exponential function, the graph of the logarithmic function is the graph of the exponential function, but reflected about the line y = x. Properties of the graph of a Logarithmic Function f(x) = log a x: 1. The domain is the set of all positive real numbers. The range is the set of all real numbers. 2. The x-intercept of the graph is 1. There is no y-intercept. 3. The vertical asymptote is the line x = 0 (the y-axis). 4. If 0 < a < 1 then the function is decreasing. If a > 1 then the funtion is increasing. 5. The graph of f contains the points (1, 0), (a, 1), ( 1 a, 1). 3
22 ex. Let (a) Find the domain of f. f(x) = 2 log 3 (x + 1) (b) Graph f. (c) From the graph, determine the range and any asymptotes of f. (d) Find f 1, the inverse of f. 4
23 (e) Use f 1 to find the range of f. (f) Graph f 1. Solve: a) log 2 (5x 2) = 3 b) ln e 3x = 9 5
24 c) log 5 25 = 8x + 4 d) e 3x+2 = 5 e) log 4 (x 2 + x + 8) = 3 f) log 2 8 x = 3 6
25 Section 6.4: Example Answers ex. Let (b) Graph f. f(x) = 2 log 3 (x + 1) (f) Graph f 1. 1
26 2
27 Section 6.5: Properties of Logarithms Properties of Logarithms: 1. log a 1 = 0 2. log a a = 1 3. a log a M = M 4. log a a r = r 5. log a (MN) = log a M + log a N ( 6. log M ) a N = loga M log a N 7. log a M r = r log a M 8. If M = N, then log a M = log a N 9. If log a M = log a N, then M = N 10. Change of base formulas: log a M = log b M log b a ex. Find the exact value: a) ln e 5 = log M log a = ln M ln a b) log log 4 8 c) log 5 7 log
28 Write each expression as a sum and/or difference of logarithms. powers as factors. Express a) ln x2 e 2x b) ln [ (x 4) 2 x 2 1 ] 2 3, x > 4 ex. Express as a single logarithm: ( ) ( ) a) log log x 2 +2x 8 x 2 +x 6 x 2 2x 3 x+4 b) 24 log 4 4 x + log 4 (8x 2 ) log 4 8 2
29 Section 6.6: Logarithmic and Exponential Equations ex. Solve: a) 2 log 3 (x + 4) log 3 9 = 2 b) 2 2x + 2 x+2 12 = 0 c) 3 x = 14 d) 2 x+1 = 5 1 2x 1
30 e) log 2 (3x + 2) log 4 x = 3 2
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