Math Chapter 1 Review Solution

Chapter 1 Review Name: Hour: 1. If the point (-2, 1) falls on a line with slope m = -2, give 3 additional points through which the line would fall. (-1, -1), (0, -3), (1, -5) 2. In 2015, a product had a value of $3795. Over the next five years, the product is expected to decrease in value by $115 per   year. Write a linear equation that gives the dollar value V of the product in terms of the year   t. V = -115t + 3795 3. During the second and third quarters of the year, an e-commerce company had sales of $160,000 and $185,000, respectively. The growth of sales follows a linear pattern. Estimate the sales during the fourth quarter. 185000 - 160000 = 25000   185000 + 25000 = $210,000 Evaluate the function at each specified value and simplify. 4. f(x) = 3 / (2x - 5) a) f(1) = 3 / (2(1) - 5) = -1 b) f(5/2) = undefined c) f(-x) = 3 / (2(-x) - 5) = 3 / (-2x - 5) d) f(x+1) = 3 / (2(x+1) - 5) = 3 / (2x - 3) 5. f(x) = { 2x + 1 if x ≤ -1, x² + 2 if x > -1 } a) f(0) = (0)² + 2 = 2 b) f(-1) = 2(-1) + 1 = -1 c) f(-5) = 2(-5) + 1 = -9 d) f(6) = (6)² + 2 = 38 Find the domain of the function in interval notation. 6. f(x) = x² / (x² + 3) (-∞, ∞) 7. f(x) = x / (x - 1) (-∞, 1) U (1, ∞) 8. f(x) = √(6 - x) (-∞, 6] 9. f(x) = 5 / √(x - 2) (2, ∞) 10. f(x) = √(x² - 4) (-∞, -2] U [2, ∞) 11. f(x) = √(x + 7) / (x - 5) [-7, 5) U (5, ∞) 12. A large rectangular pen is being fenced off at a local zoo for horses. It is to be divided in two by another fence parallel to one side of the pen. Two thousand and four hundred feet of fencing is available for use.   a) Find a function that models the area of the pen in terms of its width and determine the domain of the function.   [Image of a rectangular pen divided into two] A = x(1200 - 3/2x) A = -3/2x² + 1200x Domain: (0, 800) b) What is the maximum area that can be fenced? x = -1200 / 2(-3/2) = 400 A = 240,000 ft² Find the difference quotient and simplify your answer. Difference Quotient: (f(x+h) - f(x)) / h 13. f(x) = 2x² + 3x - 1 f(x+h) = 2(x+h)² + 3(x+h) - 1 = 2x² + 4xh + 2h² + 3x + 3h - 1 (2x² + 4xh + 2h² + 3x + 3h - 1 - 2x² - 3x + 1) / h = (4xh + 2h² + 3h) / h = 4x + 2h + 3 14. f(x) = 5 - x² f(x+h) = 5 - (x+h)² = 5 - (x² + 2xh + h²) = -x² - 2xh - h² + 5 (-x² - 2xh - h² + 5 - 5 + x²) / h = (-2xh - h) / h = -2x - h For the following functions, a) use a calculator to sketch the graph of the function, b) determine intervals on which the function is increasing, decreasing, or constant, and c) tell whether the function is one-to-one.   15. f(x) = |x - 2| + |x| [Image of the graph of f(x) = |x - 2| + |x|] Inc: (2, ∞) Dec: (-∞, 0) Constant: (0, 2) Not One-to-One 16. f(x) = √(9 - x²) [Image of the graph of f(x) = √(9 - x²)] Inc: (-3, 0) Dec: (0, 3) Constant: none Not One-to-One 17. Sketch the graph of the function BY HAND. f(x) = { 3x + 5 if x < 0 { x - 4 if x ≥ 0 [Image of a piecewise function graph] 18. Sketch the graph of the function BY HAND. f(x) = { (x - 2)² if x ≤ 2 { x² + 3 if -2 < x ≤ 2 { x² - 2x if x > 2 [Image of a piecewise function graph] Determine algebraically whether the function is even, odd, or neither. 19. f(x) = -x² + 2x f(-x) = -(-x)² + 2(-x) = -x² - 2x neither 20. f(x) = x³ - 2x f(-x) = (-x)³ - 2(-x) = -x³ + 2x odd Consider the functions f(x) = |x| and g(x) = |(1/3)x| + 4. 21. a) Graph f(x). [Image of the graph of f(x) = |x|] b) Describe the transformations from f(x) to g(x). Horizontal Stretch of 3 Vertical Shift Up 4 c) Graph g(x). [Image of the graph of g(x) = |(1/3)x| + 4] d) Use function notation to write g in terms of f. g(x) = f((1/3)x) + 4 Consider the functions f(x) = √x and g(x) = 2√(-x) - 1. 22. a) Graph f(x). [Image of the graph of f(x) = √x] b) Describe the transformations from f(x) to g(x). Reflection over y-axis Vertical Stretch of 2 Vertical Shift of 1 down c) Graph 1   31. Consider the functions f(x) = x² and g(x) = -(x - 2)² + 5. a) Graph f(x) [Image of a parabola opening upwards] b) Describe the transformations from f(x) to g(x). Horizontal Shift Right 2 Reflection over x-axis Vertical Shift Up 5 c) Graph g(x). [Image of a parabola opening downwards, shifted right 2 units and up 5 units] d) Use function notation to write g in terms of f. g(x) = -f(x - 2) + 5 The graph of f is given. Use it to graph each of the following functions. 24. y = -f(2x) + 1 [Image of the graph of y = -f(2x) + 1] 25. y = 2f(x + 1) [Image of the graph of y = 2f(x + 1)] Let f(x) = 3 - 2x and g(x) = -2x² + 1. Evaluate the following. 26. (f + g)(x) -2x² - 2x + 4 27. (f * g)(x) (3 - 2x)(-2x² + 1) -6x² + 3 + 4x³ - 2x 4x³ - 6x² - 2x + 3 28. (g o f)(x) -2(3 - 2x)² + 1 -2(9 - 12x + 4x²) + 1 -18 + 24x - 8x² + 1 -8x² + 24x - 17 29. (f - g)(-5) f(-5) - g(-5) 13 - (-49) 62 30. (f o f)(5) f(5) = -7 f(-7) = 3 - 2(-7) = 17 Find the inverse of the function algebraically. 32. f(x) = √(x + 5) x = √(y + 5) x² = y + 5 y = x² - 5 f⁻¹(x) = x² - 5 33. f(x) = -x³ - 2 x = -y³ - 2 x + 2 = -y³ -x - 2 = y³ f⁻¹(x) = ∛(-x - 2)