Chapter 1 reveal

Chapter 1 Review: Pre-Calculus 342

No Calculator Allowed

Problem 1: Inequality Notation vs. Interval Notation

  • Inequality Notation vs. Interval Notation Table

    • a) -15 < x < 4

      • Interval Notation: [-15, 4)

    • b) [-6, 0) U [2, ∞)

    • c) x < 3 or x > 8

      • Interval Notation: (-∞, 3) U (8, ∞)

    • d) -7 < x ≤ -1

      • Interval Notation: (-7, -1]

Problem 2: Descriptions of Relations

  • For each relation, write letters of applicable descriptions:

    • a) f(x) = (x + 4)² + 2

      • Applicable Descriptions: A.

    • b) f(x) = 13|x| + 2

      • Applicable Descriptions: A, B, C.

    • c) [relation not specified]

      • Applicable Descriptions: A, C, D.

    • d) [relation not specified]

      • Applicable Descriptions: A, C, D.

    • e) [relation not specified]

      • Applicable Descriptions: none.

    • f) [relation not specified]

      • Applicable Descriptions: A, C, D.

    • g) [relation not specified]

      • Applicable Descriptions: A, C, D.

Problem 3: End Behavior of Functions

  • Describe end behavior:

    • a) As x → -∞, f(x) → ? (specific behavior not provided in text)

    • b) As x → +∞, f(x) → ? (specific behavior not provided in text)

Problem 4: Limits

  • Use graph to find limits:

    • a) lim f(x) as x → 3- = 7

    • b) lim f(x) as x → 3+ = 5

    • c) lim f(x) as x → -5- = 2

    • d) lim f(x) as x → -5+ = -10

    • e) lim f(x) as x → -1- = 3

    • f) lim f(x) as x → -1+ = -6

    • g) f(-1) = ? (value not provided)

    • h) Identify discontinuities:

      • Type at x = -5: jump discontinuity.

      • Type at x = -4: removable discontinuity.

      • Type at x = -1: removable discontinuity.

      • Type at x = 3: infinite discontinuity.

Problem 5: Domain and Range

  • For each function:

    • a)

      • i) Domain: All real numbers except -4 ≤ x < -2

        • Interval: (-∞, -4) U [-2, ∞)

      • ii) Range: Real numbers ≥ -6

        • Interval: [-6, ∞)

      • iii) Behavior:

        • Increasing: -2 < x < 1

          • Interval: (-2, 1)

        • Decreasing: x < -4 and 1 < x < 4

          • Intervals: (-∞, -4) and (1, 4)

        • Constant: x = 4

          • Interval: (4, ∞)

    • b)

      • i) Domain: All real numbers

        • Interval: (-∞, ∞)

      • ii) Range: All real numbers except 2

        • Interval: (-∞, 2] U (4, ∞)

      • iii) Behavior:

        • Increasing: All real numbers

          • Interval: (-∞, ∞)

        • Decreasing: never

        • Constant: never

Problem 6: Finding Zeros

  • For given functions, find zero(s) and domain:

    • a) f(x) = 6x³ + 7x² - 3x

      • i) Zeros: (0, 0), (1, 0), (2, 0)

    • b) h(x) = 3x + 18

      • i) Domain: All real numbers

        • Interval: (-∞, ∞)

    • c) h(x) = √(x² - 10)

      • i) Domain: x < -√10 or x > √10

Problem 7: Graph Functions

  • Accurately graph the following:

    • a) f(x) = √(x) + a

    • b) f(x) = 1/x + b

    • c) f(x) = [function not specified] ...

Problem 8: Transformations of Graphs

  • Sketch graphs based on transformations:

    • a) -1/(f)

    • b) f(-x) - 4

    • c) |f(x)|

    • d) f(2x) + 3

    • e) -2f(x - 2) + 1

    • f) |f(x)|

Problem 9: Piecewise Functions Graphing

  • Graph the following piecewise functions:

    • a) f(x) = {

      • x - 3 if x < -1

      • 3(x + 2) - 2 if -3 ≤ x < 0

      • [additional functions not specified]

    • b) f(x) = {

      • {function expressions provided}

      • [additional functions not specified]

Problem 10: Function Operations and Simplification

  • Match operations to simplified forms:

    • (f ° g)(x) = D (f(g))

    • (f + g)(x) = B

    • (f - g)(x) = A

    • (fg)(x) = E

Problem 11: Function Operation Results

  • For given functions, find results:

    • a) (f + g)(x) = -2x² + x - 8

    • b) (g - f)(x) = 2x² + x - 14

    • c) (f ° g)(x) = -2x³ + 22x² + 3x - 33

Problem 12: Graphs of Inverse Functions

  • Sketch inverse functions based on function graphs shown:

    • a) [original function not specified]

    • b) [original function not specified]

    • c) [original function not specified]

Problem 13: Inverse Function Equations

  • Write equations of inverse functions:

    • a) f(x) = √(3x - 12)

    • b) [original function not specified]

Problem 14: Composite Functions

  • Find two functions such that h(x) = (f ° g)(x):

    • a) h(x) = (2x + 1)²

    • b) h(x) = √(9x - 14)

robot