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Exercises 1—14, to establish a big-Orelationship, find witnessesCandksuch that
Determine whether each of these functions is
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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
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- In Exercises 3–10, differentiate the expression with respect to x, assuming that y is implicitly a function of x.arrow_forwardIn Exercises 13-14, find the domain of each function. 13. f(x) 3 (х +2)(х — 2) 14. g(x) (х + 2)(х — 2) In Exercises 15–22, let f(x) = x? – 3x + 8 and g(x) = -2x – 5.arrow_forwardIn Exercises 7–10,find the two x-intercepts of the function f andshow that f '(x) = 0 at some point between the twox-intercepts. f (x) = x2 − x − 2arrow_forward
- Use Definition 0.10 to show that each pair of functions in Exercises 67–70 are inverses of each other. 1 2 67. f(x) =2 – 3x and g(x) = -x+ 3 68. f(x) = x² restricted to [0, 0) and g(x) = V 69. f(x) = and g(x) = 1+x 1-x 1 1 70. f(x) = and g(x) 2x 2xarrow_forwardDefine f(x) = 4x - 3 x² + 1 -e-x. a) Evaluate f(0.9) and f(1.0) correct to 2 decimal places. f(0.9) = f(1.0) =arrow_forwardSometimes a derivative contains negative exponents, andsimplification requires that all exponents be positive.Write each of the expressions in Exercises 46–49 withoutnegative exponents.arrow_forward
- Define S if x 1 x² – 5 f(x) = if x > 1 Calculate f(2).arrow_forwardIn Exercises 51–54, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be?arrow_forwardIn Exercises 16–22, show that the two functions are inverses of each other. 2 16. f(x) = 3x + 2 and g(x) = 3arrow_forward
- In Exercises 83–85, you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Per-form the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where ƒ′ = 0. (In some exercises, you may have to use the numerical equation solver to ap-proximate a solution.) You may want to plot ƒ′ as well. c. Find the interior points where ƒ′ does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function’s absolute extreme values on the interval and identify where they occur. 83. ƒ(x) = x4 - 8x2 + 4x + 2, [-20/25, 64/25] 84. ƒ(x) = -x4 + 4x3 - 4x + 1, [-3/4, 3] 85. ƒ(x) = x^(2/3)(3 - x), [-2, 2]arrow_forwardIn Exercises 104–105, express the given function h as a composition of two functions f and g so that h(x) = (f• g)(x). 104. h(x) = (x² + 2x – 1)* 105. h(x) = V7x + 4 %3! %3!arrow_forwardUse graphs to determine if each function f in Exercises 45–48 is continuous at the given point x = c. [2 – x, if x rational x², if x irrational, 45. f(x) c = 2 x² – 3, if x rational 46. f(x) = { 3x +1, if x irrational, c = 0 [2 – x, if x rational 47. f(x) = { x², if x irrational, c = 1 x² – 3, if x rational 3x +1, if x irrational, 48. f(x) : c = 4arrow_forward
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