Unformatted text preview: m(t) = 1 (24) = 24 · 2−t/25 2t/25 There was some talk about the number e More on that later with Logarithms and natural Logarithms. Section 1.6: Inverse Function and Logarithms
Not all functions have inverses. Recall that a function is deﬁned as a rule f that assigns to each element x in a set A exactly one element, called f (x) in a set B . This requirement that a function have exactly one element imposes a condition on whether or not a function f has an inverse function. A function f is called a onetoone function if it never takes on the same value twice, that is f (x1 ) = f (x2 ) whenever x1 = x2 . You can check whether or not a function is onetoone from the graph by using the horizontal line test.
Chapter 1 Review 7 Deﬁnition Horizontal Line Test than once. A function is onetoone if and only if no horizontal line intersects its graph more Deﬁnition Let f be a onetoone function with domain A and range B . Then its inverse function f −1 has domain B and range A and is deﬁned by: for any y in B . f −1 (y ) = x ←→ f (x) = y The letter x is usually reserved for the independent variable, so we can rewrite our deﬁnition as: when we are talking about the inverse function. Cancellation Equations f −1 (f (x)) = x for every x in A f (f −1 (x)) = x for every x in B f −1 (x) = y ←→ f (y ) = x How to ﬁnd the inverse function of a onetoone function Step 1 Write y = f (x). Step 2 Solve this equation for x in terms of y (if possible). Step 3 To express f −1 as a function of x, interchange x and y . The resulting equation is y = f −1 (x) Inverse from a Graph Since f (a) = b iﬀ f −1 (b) = a, the point (a, b) is on the graph of f iﬀ the point (b, a) is on the graph of f −1 . But we get the point (b, a) from reﬂecting about the line y = x: Technique The graph of f −1 is obtained by reﬂecting the graph of f about the line y = x. Logarithmic Functions
If a > 0 and a = 1, the exponential function is either increasing or decreasing and so it is onetoone by the Horizontal Line Test. Therefore, it has an inverse, f −1 which is called the logarithmic function with base a and is denoted loga . Using our deﬁnition of inverse functions, we have loga (x) = y ←→ ay = x. So, if x > 0, then loga x is the exponent to which the base a must be raised to give x: log10 0.0001 = −4 since 10−4 = 0.0001 Inverse from a Graph Since f (a) = b iﬀ f −1 (b) = a, the point (a, b) is on the graph of f iﬀ the point (b, a) is on the graph of f −1 . But we get the point (b, a) from reﬂecting about the line y = x: Technique The graph of f −1 is obtained by reﬂecting the graph of f about the line y = x. Logarithmic Functions
If a > 0 and a = 1, the exponential function is either increasing or decreasing and so it is onetoone by the Horizontal Line Test. Therefore, it has an inverse, f −1 which is called the logarithmic function with base a and is denoted loga . Using our deﬁnition of inverse functions, we have loga (x) = y ←→ ay = x. So, if x > 0, then loga x is the exponent to which the base a must be raised to give x: log10 0.0001 = −4 since 10−4 = 0.0001 The cancellation equations can be written for the logarithmic and exponential functions as: loga (ax ) = x for every x ∈ (−∞, ∞) aloga (x) = x for every x ∈ (0, ∞) Chapter 1 Review 8 Laws of Logarithms If x and y are positive numbers, then loga (xy ) x loga y loga (xr ) = = loga x + loga y loga x − loga y = r loga x where r is any real number Natural Logarithms
The natural logarithm has as base the number e. It is usually written loge x = ln x. The deﬁning properties of the natural logarithm are: Cancellation equations: ln x = y ←→ ey = x ln(ex ) = x eln x = x If we set x = 1, we see that ln e = 1. x ∈ (−∞, ∞) x>0 Theorem For any positive number a(a = 1), we have loga x = ln x . ln a Understand the proof! Example: 90 Sr decay We looked at the example of radioactive decay, and found how the mass of Strontium 90 decayed as a function of time t (years). Now we can ﬁnd the inverse function and interpret what it means for this case. The equation we determined before was: Take the natural logarithm of both sides: m = f (t) = 24 · 2−t/25 ln m = ln(24 · 2−t/25 ) t ln m = ln 24 + ln 2−t/25 t ln m = ln 24 − ln 2 25 24 Theorem For any positive number a(a = 1), we have loga x = Understand the proof! ln x . ln a Example: 90 Sr decay We looked at the example of radioactive decay, and found how the mass of Strontium 90 decayed as a function of time t (years). Now we can ﬁnd the inverse function and interpret what it means for this case. The equation we determined before was: Take the natural logarithm of both sides: m = f (t) = 24 · 2−t/25 ln m = ln(24 · 2−t/25 ) 25 24 ln ln 2 m which gives the time it takes for the mass to decay to m milligrams. t = f −1 (m) = So the inverse function is ln m = ln 24 + ln 2−t/25 t ln 2 ln m = ln 24 − 25 t 24 ln 2 = ln 24 − ln m = ln 25 m 24 25 ln t= ln 2 m ...
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 Fall '08
 JUNGHENN
 Calculus, Inverse function, Natural logarithm, Logarithm, horizontal line test

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