Section1.6

Section1.6 - m(t) = 1 (24) = 24 · 2−t/25 2t/25 There was...

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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 one-to-one 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 one-to-one from the graph by using the horizontal line test. Chapter 1 Review 7 Definition Horizontal Line Test than once. A function is one-to-one if and only if no horizontal line intersects its graph more Definition Let f be a one-to-one 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 one-to-one 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 one-to-one 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 one-to-one 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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