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1 Section.7 Definition of Inverse Function Let f and g be two functions such that f(g(x)) = x for every x in the gomain of g and g(f(x)) = x for every x in the gomain of f Under these conditions, the function g is the inverse function of the function f. The function g is denoted by f (read f-inverse ). So, f(f (x)) = x and f (f(x)) = x The domain of f must be equal to the range of f, and the range of f must be equal to the domain of f. Note: f (x) does not mean f(x). Show that f and g are inverse functions. f(x) = x 8 g(x) = p 8x Graph the inverse function. p f(g(x)) = f 8x = = 8x 8 = x p 8x x g(f(x)) = g 8 x = s8 8 = p x = x 8
2 Horizontal Line Test for Inverse Functions A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point. One-to-One Functions A function is one-to-one (-) if each value of the dependent variable corresponds to exactly one value of the independent variable. That is, f is - if and only if x 6= x implies f(x ) 6= f(x ). Note Sometimes we cheat. For instance, f(x) =x, for x 0, has the inverse function f (x) = p x. We call this restricting the domain. Finding an Inverse Function. Use the Horizontal Line Test to decide whether f has an inverse function.. In the equation for f(x), replace f(x) byy.. Interchange the roles of x and y, and solve for y. 4. Replace y by f (x) in the new equation. 5. Verify that f and f and inverse functions of each other by showing that the domain of f is equal to the range of f, the range of f is equal to the domain of f, and f f (x) = x and f (f(x)) = x. Find the inverse of the function f(x) = p x Find the inverse of the function f(x) = p x y = p x x = p y x = y x + = y f (x) = x + f(x) = 8x 4 x +6
3 f(x) = 8x 4 x +6 y = 8x 4 x +6 x = 8y 4 y +6 x(y + 6) = 8y 4 yx +6x = 8y 4 yx 8y = 6x 4 y(x 8) = 6x 4 y = 6x 4 x 8 f (x) = 6x +4 x 8 Homework.7-6, 5-8, 9,,, 4, 7, 9, 5, 9-4, 49, 55, 6, 65, 75, 87, 9
4 Section 5. Definition of Exponential Function The exponential function f with base a is denoted by f(x) =a x where a>0, a 6=, and x is any real number. f(x) = x f(x) = x f(x) = 0.7 x What is a 0? a 0 =. Graphing f(x) = x and g(x) = x. f(x) = x x 0 x x g(x) = x Graph of y = a x, a> Domain: (, ) Range: (0, )
5 y-intercept: (0,) Increasing x-axis is a horizontal asymptote (a x! 0 as x! ). Continuous Graph of y = a x, a> Domain: (, ) Range: (0, ) y-intercept: (0,) Decreasing x-axis is a horizontal asymptote (a x! 0 as x!). Continuous Recall the transformation of graphs. h(x) = x h(x) = x h(x) = x + Note: An exponential function is one-to-one. So, we can use the one-toone property. For a>0 and a 6=, a x = a y if and only if x = y Solve for x. = x and = 4 x = x 5 = x 5 = x 8 = x
6 = 4 x 5 = ( ) x 5 = ( x) 5 = ( x) 5 = 4 x = x = x The natural base e e The function given by f(x) =e x is called the natural exponential function. How do we get e? + m m! e as m! Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas.. For n compoundings per year: A = P a + r n nt. For continuous compounding: A = Pe rt annually ) n = semiannually ) n = quarterly ) n =4 monthly ) n = daily ) n = 65 continuously ) use A = Pe rt A total of $,58,497.5 is invested at an annual interest rate of 4%. Find the balance after 5 years if it is compounded. semiannually. mounthly. continuously
7 First observe that P =, 58, r = 0.04 t = 5 For semiannually, we have n =. So, For mounthly, we have n =. So, A = P a + r nt n =, 58, =, 79, A = P a + r nt n =, 58, =, 90, For continuously, we use A = Pe rt. So, A = Pe rt =, 58, 497.5e (0.04)(5) =, 9,.50 homework 5. -6, -6,, -8, 5, 5, 55, 57, 59, 77, 78 ()(5) ()(5) 4
8 Section 5. Definition of Logarithmic Function with Base a For x>0, a>0, and a 6=, The function given by y = log a x if and only if x = a y f(x) =log a x is called the Definition of Logarithmic Function with Base a. log = log 5 =5 log 8 = log 4 =4 log = log =6. The logarithmic function with base 0 is called the common loragithmic function. It is denoted by log 0, or simply by log. Properties of Logarithms log a = 0 because a 0 = log a a = because a = a log a a x = x and a log a x = x 4 If log a x = log a y, then x = y log ( 4x) = log (9) log (x ) = log (9) Graphing f(x) = x and g(x) = log x.
9 Graph of y = log a x, a> Domain: (0, ) Range: (, ) x-intercept: (,0) Increasing One-to-one, therefore it has an inverse function (which is y = a x ) y-axis is a vertical asymptote (log a x! as x! 0 + ). Continuous reflection of graph of y = a x about the line y = x. The Natural Logarithmic Function The function defined by f(x) = log e x =lnx is called the the natural logarithmic function. Properties of Natural Logarithms
10 ln = 0 because e 0 = lne = because e = e lne x = x and e ln ex = x 4 If lnx =lny, then x = y What is the domain of f(x) = log 4 (x 4)? We want x 4 > 0. So, all real numbers such that x > 4. Alternatively, {x : x>4}. homework 5. -6, -8, -6, 7, 4, 45-50, 7, 7, 75, 77, 85, 87, 89, 9
11 Section 5. Change-of-Base Formula Let a, b, and x be positive real numbers such that a 6= and b 6=. Then log a x can be converted to a di erent base as follows. Base a Base 0 Base e log a x = log b x log b a log a x = log x log a log a x = ln x ln a log 5 log p 7 p log 7 log Properties of Logarithms Let a be a positive number such that a 6=, and let n be a real number. If u and v are positive real numbers, the following properties are true.. Product Property: log a (uv) = log a u + log a v. Quotient Property: log a u v = log a u log a v. Power Property: Expand the expression log a u n = n log a u ln kx 5. ln kx 5 ) ln(kx ) ln(5) ) ln(k)+ln(x ) ln(5) ) ln(k)+ln(x) ln(5)
12 Write log 5(x + ) + log 5(x ) as a logarithm of a single expression log 5(x + ) + log 5(x ) ) log 5 (x + ) + log5 (x ) ) log 5 p x + + log 5 p x ) log p 5 x + p x p ) log 5 (x + )(x ) p ) log 5 x homework 5. -6, 7,, 5,, 9,, 5, 7, 9, 5, 59, 6, 65, 75, 77, 79, 8, 8
13 Section 5.4 Solve the exponential equation 5(.) x + 94 = 00. 5(.) x + 94 = 00 5(.) x = 6 (.) x = 6 5 (.) x =. log(.) x = log. (x ) log(.) = log. x = log. log(.) = x = x = Check answer... Solve the exponential equation e x = e 5x. e x = e 5x ln e x = lne 5x x = 5x x = x = Check answer... Solve the exponential equation 6 5x = 000.
14 6 5x = 000 ln 6 5x = ln 000 5x ln 6 = ln 000 x = ln 000 5ln6 This is an exact answer. Check answer... Solve the logarithmic equation log 6 (x + 4) =. Check answer... Solve the logarithmic equation log 6 (x + 4) = 6 log 6 (x+4) = 6 x +4 = 6 x + 4 = 6 x = x = 6 log x + log (x + ) = log 4. log x + log (x + ) = log 4 log x(x + ) = log 4 log x(x+) = log 4 x(x + ) = 4 x +x 4 = 0 (x 4)(x + 6) = 0
15 So, x = 4 and x = 6, but when x = 6, log ( 6) is undefined. Thus, x = 6 is not a solution to this logarithmic equation. Therefore, x = 4 is the only solution. Solve the logarithmic equation ln p x 8=5. ln p x 8 = 5 e ln p x 8 = e 5 p x 8 = e 5 p x 8 = e 0 x 8 = e 0 x = e 0 +8 This is an exact answer. Check answer... homework 5.4-4, 7, 9,, 9,,, 9, 5, 55, 65, 69, 89, 9, 99, 0, 0,, 7
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