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CHAPTER 1

CHAPTER 1. A function f is a rule that assigns to each element x in a set A exactly one element, called f (x), in a set B. The set A is called the domain of the function. The range of the function is the set of all possible values of f (x) as x

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CHAPTER 1

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  1. CHAPTER 1 A function f is a rule that assigns to each element x in a set A exactly one element, called f (x), in a set B.

  2. The set A is called the domain of the function. The range of the function is the set of all possible values of f (x) as x varies throughout the domain.

  3. Increasing and Decreasing Functions A function f f is called increasing on an interval I if f(x1) < f(x2) whenever x1 < x2in I.

  4. Increasing and Decreasing Functions A function f f is called decreasing on an interval I if f(x1) > f(x2) whenever x1 < x2in I.

  5. 1.5 Exponential Functions An exponential function is a function is a function of the form f(x) = axwhere a is a positive constant. If x = n, a is a positive integer, then an = a .a . a… . … . a(n factors)

  6. If x = 0,then a0=1, and if x = -n where n is a positive integer, then a-n=1/an . If x is a rational number, x=p/q where p and q are integers and q > 0, then ax=a p/q = qa p .

  7. Laws of Exponents If a and b • are positive numbers and x and • y are any real numbers, then • ax+y = ax ay • 2. ax-y = ax /ay • 3. (ax) y = ax y • 4. (ab)x = axby

  8. Definition of EIn the family of exponential functions f(x) = bx there is exactly one exponential Function for which the slope of the Line tangent at (0,1) is exaclty 1. This occurs for b=2.71…This Important number is denoted by e.

  9. 1.6 Inverse Functions and Logarithms Definition A function f is called a one-to-one function if it never takes on the same value twice; that is, f(x1) is not equal to f(x2) whenever x1is not equal to x2 .

  10. Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once.

  11. 1.6 Inverse Functions and Logarithms Definition Let f be a one-to-one function with domain A and range B. Then its inverse function f –1has domain B and range A and is defined by f –1(y) = x, then f(x) = y for any y in B.

  12. domain of f –1 = range of f range of f –1 = domain of f . f –1 (x) = y then f(y) = x. f –1 (f (x)) = x for every x in A f (f –1(x)) = x for every x in B.

  13. The graph of f –1is obtained by reflecting the graph of f about the line y = x.

  14. The graph of f –1is obtained by reflecting the graph of f about the line y = x. ln x = y then e y = x

  15. The graph of f –1is obtained by reflecting the graph of f about the line y = x.

  16. Laws of Logarithms • If x and y are positive numbers, • then • ln (x y) = lnx + lny • ln (x/y) = lnx - lny • ln (xr) = r lnx (where r is • any real number)

  17. lnx = y then e y = x

  18. lnx = y then e y = x ln(e x) = x x  R e ln x = x x > 0

  19. lnx = y then e y = x ln(e x) = x x  R e ln x = x x > 0 lne = 1

  20. For any positive number a ( a is not equal to 1), we have log a x = ln x / ln a

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