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)