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Lecture 5

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Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

MATH140 - Lecture 5 Notes Inverse Functions & Logs: Definition: A function f(s) is a 1:1 function if it never takes on the same value force twice Horizontal Line Test: A function is 1-to-1 if and only if no horizontal line intersects its graph more than once Example: f(x) = 5-4x^3 • Step 1: Replace f(x) for y • Step 2: Solve for x • Step 3: Interchange x for y y = 5-4x^3 4x^3 = 5-y x^3 = (5-y)/4 x = root3((5-y)/4) or x = ((5-y)/4)^(⅓) Another way of understanding if a function f(x) is one to one is to see if we can find its inverse denoted by f^-1(x). To explain the inverse in simple terms, if f takes x to y then f^-1 takes y to x. f(x) = y 㲗 f^-1(y) = x We note the following facts about inverse functions: • F(f^-1(x)) = f^-1(f(x)) = x • The graphs of f(x) and f^-1(x) are symmetric with respect to the line y = x • Domain of f = Range of f^-1 and Range of f = Domain of f^-1 Definition: Let f be a 1-to-1 function with domain A and range B. It’s inverse function f^-1 is domain B and range A and is defined by f^-1(y) = x 㲗 f(x) = y for any YEB CAUTION: Do not mistake the -1 in f^-1 for an exponent. Thus, F^-1(x) does not mean 1/f(x)… Example 2 (from textbook): Find the inverse function of f(x) = x^3 + 2 SECTION 1.6 INVERSE FUNCTIONS AND LOGARITHMS |||| V EXAMPLE 4 Find the inverse function of f▯x▯ ▯ x ▯ 2 . SOLUTION According to (5) we first write 3 y ▯ x ▯ 2 Then we solve this equation for :x 3 x ▯ y ▯ 2 x ▯ s3y ▯ 2 Finally, we interchange x and :y N In Example 4, notice hreverses the 3 effect of . The funcis the rule “Cube, y ▯ s x ▯ 2 then add 2”; is the rule “Subtract 2, then take the cube root.” Therefore the inverse function is f▯1▯x▯ ▯ sx ▯ 2 . The principle of interchanging x and y to find the inverse function also gives us the method for obtaining the graph of f▯1 from the graph of . Since f▯a▯ ▯ b if and only if ▯1 Logarithmic Functions:t ▯a, b▯is on the graph of f if and only if the point ▯b, a▯is on the graph of f▯1 . But we get the point▯b, a▯from ▯a, b▯by reflecting about the line y ▯ x 64 |||| CHAPTER 1 FUNCTIONS AND MODELS Figure 8.) If a > 0 and a ≠ 0, the exponential function f(x) = a^x is either increasing or decreasing 64 |||| CHAPTER 1 FUNCTIONS AND MODEyS (b,▯a) y then we have and so is 1:1 by the Horizontal Line Test. Therefore, it has an inverse function which is called the logthen we havection with base a shown by thf–!ollowing equation: y 6 logax ▯ y &? a ▯ x,▯b) 0 6 logax ▯ y &? a ▯ x y x x Thus, if x ▯ 0 , thenlog x is the exponent to which the base a must be raised to givex. For a y=x ▯3 y=x f example, log 10001 ▯ ▯3 As an example,Thus,ifwx ▯ 0 , thenlog a01is the exponentto whichthe base muat be raisedto give . For The cancellation equations (4), when applied to the functions f▯x▯ ▯ a x and ▯1 example, log 10001 ▯ ▯3 because 10 ▯ 0.001 . f ▯x▯ ▯ log a , become We can see the two properties based on the following calculations:he functions FIGURE 8 f▯1▯x▯ ▯ log x , become FIGURE 9
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