2024 College Algebra Final

Use the vertical line test to determine which graph represents a relation that is a function.

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• 𝑓(−4) = 127

• 𝑓(−4) = 65

• 𝑓(−4) = 27

• 𝑓(−4) = 91

Answer explanation

To find f(-4), substitute -4 into the function: f(-4) = 6(-4)^2 + 6(-4) - 7 = 6(16) - 24 - 7 = 96 - 24 - 7 = 65. Thus, f(-4) = 65.

The function 𝑓(𝑥) is represented below as a graph. Find 𝑓(4)

• 𝑓(4) = 1

• 𝑓(4) = 3

• 𝑓(4) = 2.42

• 𝑓(4) = 0

Answer explanation

To find f(4), locate x=4 on the graph. The corresponding y-value at this point is 1. Therefore, f(4) = 1 is the correct choice.

Given the piecewise function below, find 𝑓(1).

• 𝑓(1) = −8

• 𝑓(1) = 19

• 𝑓(1) = 12

• 𝑓(1) = 9

Answer explanation

To find f(1), identify the piece of the piecewise function that applies when x = 1. Evaluating that piece gives f(1) = 12, which matches the correct answer choice.

Answer explanation

To find the domain of f(x), we set the denominator x^2 + 7x - 18 ≠ 0. Factoring gives (x+9)(x-2) = 0, so x ≠ -9 and x ≠ 2. Thus, the domain is (-∞, -9) ∪ (-9, 2) ∪ (2, ∞), which matches the correct choice.

• All real numbers

Answer explanation

The function h(x) = √(-10 - x) requires the expression inside the square root to be non-negative. Thus, -10 - x ≥ 0 leads to x ≤ -10. Therefore, the domain in inequality notation is x ≤ -10.

Determine the domain and range of the function using the graph below:

• Domain: (-1,2)

  Range: (-2,2)

• Domain: [-2,2]

  Range: [-1,2]

• Domain: [-1,2]

  Range: [-2,2]

• Domain: (−∞, ∞)

  Range: (−∞, 2]

Answer explanation

The graph shows that the function is defined from -1 to 2, inclusive, giving the domain [-1, 2]. The range extends from -2 to 2, also inclusive, resulting in the range [-2, 2]. Thus, the correct choice is Domain: [-1,2] and Range: [-2,2].

Consider the function in the graph below. Determine the intervals of increase and decrease.

• Increase: (−5,1)

  Decrease:

  (−8, −5)𝑈(0, −7)

• Increase: (-5,0)

  Decrease:

  (−∞, −5)𝑈(0, ∞)

• Increase: [5,0]

  Decrease:

  (−∞, −5)𝑈(0, ∞)

Answer explanation

The function increases on the interval (-5,0) where the slope is positive, and decreases on (-∞, -5) U (0, ∞) where the slope is negative. Thus, the correct choice is Increase: (-5,0) and Decrease: (-∞, -5) U (0, ∞).

• -2

• 1/2

• 8

• 1/8

Answer explanation

To find the average rate of change of f(x) between x=2 and x=4, calculate f(4) and f(2). f(4) = 0 and f(2) = 4. The average rate of change is (f(4) - f(2)) / (4 - 2) = (0 - 4) / 2 = -2.

Answer explanation

To find (f-g)(x), subtract g(x) from f(x): f(x) - g(x) = (6x + 4) - (x^2 + 10x - 7) = -x^2 - 4x + 11. Thus, the correct answer is -x^2 - 4x + 11.