Math Pre-Calc Midterm Review Chapter 2 Solution

CHAPTER 2 - Functions Use the function f(x) = 2x^2 - 1 to evaluate the following. f(-2) f(-2) = 2(-2)^2 - 1 = 7 f(x-2) f(x-2) = 2(x-2)^2 - 1 = 2x^2 - 8x + 7 (f(a+h) - f(a))/h (f(a+h) - f(a))/h = (2(a+h)^2 - 1 - (2a^2 - 1))/h = 4a + 2h Evaluate f(0), f(3), and f(-4) for each function. f(x) = 3x^2 + x f(0) = 0 f(3) = 30 f(-4) = 44 f(x) = x + 1/x f(0) = undefined f(3) = 10/3 f(-4) = -17/4 f(x) = { 3x if x ≤ 0 { (x-2)^2 if x > 2 f(0) = 0 f(3) = 1 f(-4) = -12 Sketch the graph of the function. f(x) = 4x^2 - x^4 f(x) = x^2(4-x^2) = x^2(2+x)(2-x) f(x) = { 4 if x ≤ -2 { -x^2 if -2 < x ≤ 2 { -x+6 if x > 2 (Graph of a piecewise function with a parabola and a line) The graph of a function g is given. a) Find g(-3), g(0), and g(4) g(-3) = -2 g(0) = 1 g(4) = 5 b) Identify the domain and range of g. Domain: [-5, 4] Range: [-2, 4] 11. Determine the intervals on which the function is increasing and decreasing. Use interval notation. Increase: (-3, 0) Decrease: (-∞, -3) U (0, ∞) 12. A function f(x) = 3x - 2. Determine the average rate of change of the function between the given values. x = 2, x = 3 f(2) = 3(2) - 2 = 4 (2, 4) f(3) = 3(3) - 2 = 7 (3, 7) Average rate of change: (7-4)/(3-2) = 3 13. graph (3, 2) and (-1, 0) Average rate of change: (0-2)/(-1-3) = 1/2 Suppose the graph of f is given. Describe how the graph of g could be obtained from f. 14. g(x) = f(x) - 5 Down 5 15. g(x) = f(x-5) + 2 Right 5, Up 2 16. g(x) = 3f(x) Vertical Stretch of 3 17. g(x) = f(2x) Horizontal Shrink of 1/2, Reflect over x-axis 18. g(x) = (1/2)f(x+3) Left 3, Vertical Shrink of 1/2 19. g(x) = -f(-1/3x) + 1 Horizontal Stretch of 3, Reflect over x-axis, Up 1 20. The graph of f is given. Sketch the graph of y = -f(x+1) - 3 Left 1 Reflect over x-axis Down 3 Write the equation for the graph of f with the given transformations. f(x) = x³; shift left 3 units, reflect over the x-axis, and shift down 1 unit. f(x) = -(x+3)³ - 1 f(x) = |x|; shift right 2 units, stretch vertically by a factor of 3, and shift up 2 units. f(x) = 3|x-2| + 2 f(x) = √x; reflect over the y-axis and stretch horizontally by a factor of 2. f(x) = √(1/2x) 24. Given f(x) = 2x² + 8x + 11 a) Find the axis of symmetry. x = -8/(2*2) = -2 b) Find the vertex. f(-2) = 2(-2)² + 8(-2) + 11 = 3 (-2, 3) (min) c) Find the maximum or minimum value. y = 3 (min) 25. Express f(x) = 1 - 6x - x² in standard form. Describe the transformations on the parent function. f(x) = -(x² + 6x) + 1 f(x) = -(x² + 6x + 9) + 1 + 9 f(x) = -(x+3)² + 10 Left 3 Reflect over y-axis Up 10 Find the local maximum and minimum values of the function. graph Min: (-1, -5) Max: (4, 1) f(x) = x⁴ - 2x³ - 11x² (Use graphing calculator!) Max: (0, 0) Min: (-1.7, -13.6), (3.2, -73.3) 28. The profit P (in dollars) generated by selling x units of a certain commodity is given by P(x) = -1500 + 12x - 0.0004x². What is the maximum profit? How many units must be sold to generate it? x = -12 / (2(-0.0004)) = 15000 units P(15000) = -1500 + 12(15000) - 0.0004(15000)² = $388,500 29. If the perimeter of a rectangular garden is to be 480 ft., find the dimensions of the garden that would give the largest area. What is the maximum 1 area?   2x + 2y = 480 y = 240 - x A(x) = x(240-x) A(x) = -x² + 240x x = -240 / (2(-1)) = 120 120 by 120 14400 sq ft 30. A rancher wants to build a rectangular pig pen with an area of 100 m². a) Find a function that models the length of fencing required. f(x) = 2x + 200/x b) Find the pen dimensions that would require the minimum amount of fencing. Use Calculator: 10 x 10 m Given f(x) = x² + 2x and g(x) = x - 4, evaluate the following. Then state the domain. f + g x² + 2x + x - 4 2x² + 2x - 4 Domain: (-∞, ∞) f - g x² + 2x - (x - 4) x² + 2x - x + 4 x² + x + 4 Domain: (-∞, ∞) fg (x² + 2x)(x - 4) x³ - 4x² + 2x² - 8x x³ - 2x² - 8x Domain: (-∞, ∞) f/g (x² + 2x) / (x - 4) Domain: (-∞, 4) U (4, ∞) g/f (x - 4) / (x² + 2x) Domain: (-∞, -2) U (-2, 0) U (0, ∞) g² (x - 4)² x² - 8x + 16 Domain: (-∞, ∞) Given f(x) = x + 1 and g(x) = √(x - 1), evaluate the following. Then state the domain. (f ∘ g)(x) 1 + √(x - 1) Domain: [1, ∞) (f ∘ g)(2) g(2) = √(2 - 1) = 1 f(1) = 1 + 1 = 2 (f ∘ f)(-3) f(-3) = -3 + 1 = -2 f(-2) = -2 + 1 = -1 Draw the graph of f and use it to determine if the function is one-to-one. f(x) = x³ - x No, not one-to-one Does not pass VLT and HLT f(x) = x + 3 Yes, one-to-one Passes VLT and HLT Find the inverse of function f. f(x) = 3 - 5x y = 3 - 5x x = 3 - 5y y = (3 - x) / -5 f(x) = 5 - 4x³ x = 5 - 4y³ x - 5 = -4y³ (x - 5) / -4 = y³ y = ∛((x - 5) / -4) f(x) = √(5x + 2) x = √(5y + 2) x² = 5y + 2 x² - 2 = 5y y = (x² - 2) / 5