5. Composite Functions Example: Given f ( x ) = x 2 – 3, and g ( x ) = x + 2, find x 2 - 3 is substituted into each x in g ( x ). f ( x ) is substituted into each x in g ( x ).
6. One-to-One Functions For a function to be one-to-one, it must not only pass the vertical line test, but also the horizontal line test. A function is a one-to-one function if each value in the range corresponds with exactly one value in the domain. x y Function x y Not a one-to-one function x y One-to one function
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9. Inverse Functions Replace f ( x ) with y . Interchange x and y . Solve for y . Replace y with f –1 ( x ) . Example continued:
17. Exponential Functions For all positive numbers a , where a 1, y = log a x means x = a y . y = log a x logarithm (exponent) base number means x = a y number base exponent
20. Logarithmic Graphs Domain: { x|x > 0} Graph the function f ( x ) = log 10 x. Notice that the graph passes through the point (1,0). Example: Range:
21. Exponential vs. Logarithmic Graphs Exponential Function Logarithmic Function y = a x ( a > 0, a 1) y = log a x ( a > 0, a 1) Domain: Range: Points on Graph: x becomes y y becomes x
22. Exponential vs. Logarithmic Graphs Notice that the two graphs are inverse functions. f ( x ) f - 1 ( x ) f ( x ) = log 10 x f ( x ) = 10 x
30. Common Logarithms The common logarithm of a positive real number is the exponent to which the base 10 is raised to obtain the number. If log N = L , then 10 L = N. The antilogarithm is the same thing as the inverse logarithm. If log N = L , then N = antilog L . log 962 = 2.98318 Number Exponent antilog 2.98318 = 962 Number Exponent Example :
34. Solving Equations Example : Product Rule Property 6d. Check: Stop! Logs of negative numbers are not real numbers. True
35. § 9.7 Natural Exponential and Natural Logarithmic Functions
36. Definitions The natural exponent function is f ( x ) = e x where e 2.71823. Natural logarithms are logarithms to the base e. Natural logarithms are indicated by the letters ln. log e x = ln x Example : ln 1 = 0 ( e 0 = 1) ln e = 1 ( e 1 = e )
37. Change of Base Formula For any logarithm bases a and b , and positive number x, Change of Base Formula This is very useful because common logs or natural log can be found using a calculator. Example : Note that the natural log could have also been used.
38. Properties Notice that these are the same properties as those for the common logarithms. Properties for Natural Logarithms Product Rule Power Rule Quotient Rule Additional Properties for Natural Logarithms and Natural Exponential Expressions Property 7 Property 8
39. Solving Equations Example : Solve the following equation. Product Rule Simplify Property 6d Solve for x . Check solutions in original equation. (You will notice that only the positive 7 yields a true statement.)
40. Applications In 2000, a lake had 300 trout. The growth in the number of trout is estimated by the function g ( t ) = 300 e 0.07 t where t is the number of years after 2000. How many trout will be in the lake in a) 2003? b) 2010? In the year 2000, t = 0 . (Notice that f (0) =300 e 0.07(0) = 300 e 0 = 300, the original number of trout.) In the year 2003, t = 3. g (3) = 300 e 0.07(3) = 300 e 0.21 = 300(1.2337) 370 trout in 2003 . In the year 2010, t = 10. g (10) = 300 e 0.07(10) = 300 e 0.70 = 300(2.0138) 604 trout in 2010 . Example: