University:
Central Michigan University
Course:
MTH 130 | Pre-Calculus Mathematics
Academic year:

2023
Views:

250

Pages:

6
Author:

Genevieve Huff

Functions Review Given f(x) = -x² + 3x - 4, evaluate the following: f(-2) = -(-2)² + 3(-2) - 4 = -4 - 6 - 4 = -14 f(4) = -(4)² + 3(4) - 4 = -16 + 12 - 4 = -8 f(-x) = -(-x)² + 3(-x) - 4 = -x² - 3x - 4 f(x + 1) = -(x + 1)² + 3(x + 1) - 4 = -x² - 2x - 1 + 3x + 3 - 4 = -x² + x - 2 -1/2 f(x) = -1/2 (-x² + 3x - 4) = 1/2 x² - 3/2 x + 2 f(6) - f(x) = -(6)² + 3(6) - 4 - (-x² + 3x - 4) = -36 + 18 - 4 + x² - 3x + 4 = x² - 3x - 18 f(3x) = -(3x)² + 3(3x) - 4 = -9x² + 9x - 4 f(x + h) = -(x + h)² + 3(x + h) - 4 = -x² - 2xh - h² + 3x + 3h - 4 Given f(x) = (2x - 5) / (x - 6), evaluate the following: f(0) = (2(0) - 5) / (0 - 6) = -5 / -6 = 5/6 f(-x) = (2(-x) - 5) / (-x - 6) = (-2x - 5) / (-x - 6) = (2x + 5) / (x + 6) f(6) = (2(6) - 5) / (6 - 6) = 7/0 = undefined f(x - 2) = (2(x - 2) - 5) / (x - 2 - 6) = (2x - 4 - 5) / (x - 8) = (2x - 9) / (x - 8) Evaluate the difference quotient for the function. f(x) = 4/x (f(x + h) - f(x)) / h = (4/(x + h) - 4/x) / h = (-4h) / (x(x + h)) * (1/h) = -4 / (x(x + h)) f(x) = 2 - x² (f(x + h) - f(x)) / h = (2 - (x + h)²) - (2 - x²) / h = (-2xh - h²) / h = -2x - h f(x) = x² + 2x - 3 (f(x + h) - f(x)) / h = (x² + 2xh + h² + 2x + 2h - 3 - (x² + 2x - 3)) / h = (2xh + h² + 2h) / h = 2x + h + 2 f(x) = -3/x (f(x + h) - f(x)) / h = (-3 / (x + h) - (-3/x)) / h = (3x - 3(x + h)) / (x(x + h)) * (1/h) = (3h) / (x(x + h)) * (1/h) = 3 / (x(x + h)) Image Analysis Text Content: 17. Rewrite the interval (-∞, 4] using inequalities. x ≤ 4 18. Rewrite x ≥ -3 in interval notation. [-3, ∞) 19. Rewrite x ≠ -9 in interval notation. (-∞, -9) U (-9, ∞) 20. Rewrite 2 < x < 21 in interval notation. (2, 21) 22. Rewrite "all reals" in interval notation. (-∞, ∞) Identify the domain and range, and the open intervals where the function is increasing, decreasing, or constant. Please leave your answers in interval notation. 23. Domain: (-5, -2) U (-2, 2) U (2, 5) Range: (-∞, -1) U (0, ∞) Increasing: (0, 2) U (2, 5) Decreasing: (-5, -2) U (-2, 0) Constant: None 24. Domain: (-∞, -1) U (1, 5) U (5, ∞) Range: (-∞, 1) U (1, 2) U (2, ∞) Increasing: (-∞, -1) U (-1, ∞) Decreasing: None Constant: None 25. Domain: (-∞, ∞) Range: (-∞, 4] Increasing: (-∞, -1) U (0, 2) Decreasing: (-1, 0) Constant: None 26. Domain: (-∞, 0] Range: [1, ∞) Increasing: None Decreasing: (-∞, 0) Constant: None Visual Elements: The image is a handwritten notebook page. The text is divided into sections with headings and subheadings. Graphs of functions are included. The overall layout is clear and organized.

Math Functions Review Solution Key

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