Mathematical Reasoning Algebra and Functions Flashcards

A function is defined as f(x) = (x² − 9) / (x − 3) for x ≠ 3. Which of the following best describes the behavior of f(x) as x approaches 3?

Answer: f(x) approaches 6

Factor the numerator: x² − 9 = (x − 3)(x + 3). Canceling (x − 3) gives f(x) = x + 3 for x ≠ 3. As x approaches 3, f(x) approaches 3 + 3 = 6. Even though f(3) is undefined, the limit exists and equals 6.

A company's profit P (in thousands of dollars) is modeled by P(x) = −2x² + 20x − 42, where x is the number of units sold (in hundreds). What is the maximum profit, and at how many units does it occur?

Answer: Maximum profit of $8,000 at 500 units

The vertex of a parabola ax² + bx + c occurs at x = −b/(2a). Here, x = −20/(2·(−2)) = 5. P(5) = −2(25) + 100 − 42 = −50 + 100 − 42 = 8. So maximum profit is $8,000 (since P is in thousands) at x = 5 hundreds = 500 units.

If g(x) = 3x − 1 and h(x) = x² + 2, what is the value of g(h(−3))?

Answer: 32

First compute h(−3): (−3)² + 2 = 9 + 2 = 11. Then compute g(11): 3(11) − 1 = 33 − 1 = 32. Composition of functions requires evaluating the inner function first.

Solve for x: |2x − 5| + 3 > 10

Answer: x 6

Subtract 3 from both sides: |2x − 5| > 7. This splits into two inequalities: 2x − 5 > 7 → x > 6, or 2x − 5 6.

The table below shows selected values of a linear function f(x): | x | f(x) | |----|------| | −2 | 9 | | 1 | 3 | | 4 | −3 | What is the value of x when f(x) = −9?

Answer: 7

The slope is (3 − 9)/(1 − (−2)) = −6/3 = −2. Using point-slope with (1, 3): f(x) = −2x + 5. Set −9 = −2x + 5 → −14 = −2x → x = 7.

Which of the following is equivalent to the expression (x^(1/2) · x^(2/3)) / x^(1/6)?

Answer: x

Add the exponents in the numerator: 1/2 + 2/3 = 3/6 + 4/6 = 7/6. Then subtract the denominator's exponent: 7/6 − 1/6 = 6/6 = 1. So the expression simplifies to x¹ = x.