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COLLEGE ALGEBRA

COLLEGE ALGEBRA. LIAL HORNSBY SCHNEIDER. Function Operations and Composition. 2.8. Arithmetic Operations on Functions The Difference Quotient Composition of Functions and Domain. Operations of Functions.

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COLLEGE ALGEBRA

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  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  2. Function Operations and Composition 2.8 Arithmetic Operations on Functions The Difference Quotient Composition of Functions and Domain

  3. Operations of Functions Given two functions and g,then for all values of x for which both (x) and g(x) are defined, the functions  + g, – g, g, and /g are defined as follows. Sum Difference Product Quotient

  4. Note The condition g(x) ≠ 0 in the definition of the quotient means that the domain of (/g)(x) is restricted to all values of x for which g(x) is not 0. The condition does not mean that g(x) is a function that is never 0.

  5. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following. a. Solution Since (1) = 2 and g(1) = 8, use the definition to get

  6. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following. b. Solution Since (–3) = 10 and g(–3) = –4, use the definition to get

  7. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following. c. Solution Since (5) = 26 and g(5) = 20, use the definition to get

  8. USING OPERATIONS ON FUNCTIONS Example 1 Let (x) = x2 + 1 and g(x) = 3x + 5. Find the following. d. Solution Since (0) = 1 and g(0) = 5, use the definition to get

  9. Domains y For functions  and g,the domains of  + g,  – g, and g include all real numbers in the intersections of the domains of  and g, while the domain of /g includes those real numbers in the intersection of the domains of  and g for which g(x) ≠ 0.

  10. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let a. Solution

  11. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let b. Solution

  12. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let c. Solution

  13. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let d. Solution

  14. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let e. Give the domains of the functions. Solution To find the domains of the functions, we first find the domains of  and g. The domain of  is the set of all real numbers (–, ).

  15. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let e. Give the domains of the functions. Solution Since , the domain of g includes just the real numbers that make 2x – 1 nonnegative. Solve 2x – 1  0 to get x  ½ . The domain of g is

  16. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let e. Give the domains of the functions. Solution The domains of  + g,  – g, g are the intersection of the domains of  and g,which is

  17. USING OPERATIONS OF FUNCTIONS AND DETERMINING DOMAINS Example 2 Let e. Give the domains of the functions. Solution The domains of includes those real numbers in the intersection for which that is, the domain of is

  18. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate …

  19. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 y 9 a. 5 For ( – g)(–2),although (–2) = –3, g(–2) is undefined because –2 is not in the domain of g. x 0 2 –4 –2 4

  20. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 y 9 a. 5 The domains of  and g include 1, so x 0 2 –4 –2 4

  21. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 y 9 a. 5 The graph of g includes the origin, so x 0 2 –4 –2 4 Thus, is undefined.

  22. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate b. In the table, g(–2) is undefined. Thus, (–g)(–2) is undefined.

  23. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate b.

  24. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate b. and

  25. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 If possible, use the given representations of functions  and g to evaluate c.

  26. EVALUATING COMBINATIONS OF FUNCTIONS Example 3 c.

  27. FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x2 – 3x. Find the difference quotient and simplify the expression. Solution Step 1 Find the first term in the numerator, (x + h). Replace the x in (x) with x + h.

  28. FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x – 3x. Find the difference quotient and simplify the expression. Solution Step 2 Find the entire numerator Substitute Remember this term when squaring x + h Square x + h

  29. FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x – 3x. Find the difference quotient and simplify the expression. Solution Step 2 Find the entire numerator Distributive property Combine terms.

  30. FINDING THE DIFFERENCE QUOTIENT Example 4 Let (x) = 2x – 3x. Find the difference quotient and simplify the expression. Solution Step 3 Find the quotient by dividing by h. Substitute. Factor out h. Divide.

  31. CautionNotice that (x + h) is not the same as (x) + (h). For (x) = 2x2 – 3x in Example 4. but These expressions differ by 4xh.

  32. Composition of Functions and Domain If  and g are functions, then the composite function, or composition, of g and  is defined by The domain of is the set of all numbers x in the domain of  such that (x)is in the domain of g.

  33. EVALUATING COMPOSITE FUNCTIONS Example 5 Let (x) = 2x – 1 and g(x) a. Solution First find g(2). Now find

  34. EVALUATING COMPOSITE FUNCTIONS Example 5 Let (x) = 2x – 1 and g(x) b. Solution Don’t confuse composition with multiplication

  35. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 a. Solution Multiply the numerator and denominator by x.

  36. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 a. Solution The domain of g is all real numbers except 0, which makes g(x) undefined. The domain of  is all real numbers except 3. The expression for g(x), therefore cannot equal 3; we determine the value that makes g(x) = 3 and exclude it from the domain of

  37. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 a. Solution The solution must be excluded. Multiply by x. Divide by 3.

  38. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 a. Solution Divide by 3. Therefore the domain of is the set of all real numbers except 0 and ⅓, written in interval notation as

  39. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 b. Solution Note that this is meaningless if x = 3

  40. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 b. Solution The domain of  is all real numbers except 3, and the domain of g is all real numbers except 0. The expression for (x), which is , is never zero, since the numerator is the nonzero number 6.

  41. DETERMINING COMPOSITE FUNCTIONS AND THEIR DOMAINS Example 7 b. Solution Therefore, the domain of is the set of all real numbers except 3, written

  42. SHOWING THAT Example 8 Let (x) = 4x + 1 and g(x) = 2x2 + 5x. Solution Square 4x + 1; distributive property.

  43. SHOWING THAT Example 8 Let (x) = 4x + 1 and g(x) = 2x2 + 5x. Solution Distributive property. Combine terms.

  44. SHOWING THAT Example 8 Let (x) = 4x + 1 and g(x) = 2x2 + 5x. Solution Distributive property

  45. FINDING FUNCTIONS THAT FORM A GIVEN COMPOSITE Example 9 Find functions  and g such that Solution Note the repeated quantity x2 – 5. If we choose g(x) = x2– 5 and (x) = x3– 4x + 3, then There are other pairs of functions  and g that also work.

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