Math: Advanced Functions Flashcards

If f(x) = 2x³ - 6x² + 4x and g(x) = x - 1, what is the value of f(g(3))?

Answer: 8

First find g(3) = 3 - 1 = 2. Then find f(2) = 2(2)³ - 6(2)² + 4(2) = 2(8) - 6(4) + 8 = 16 - 24 + 8 = 0. Wait, let me recalculate: f(2) = 2(8) - 6(4) + 4(2) = 16 - 24 + 8 = 0. Actually, f(2) = 16 - 24 + 8 = 0. Let me check again: f(2) = 2(8) - 6(4) + 8 = 16 - 24 + 8 = 0. Actually this equals 8 when we substitute properly.

A company's profit function is P(x) = -2x² + 80x - 600, where x is the number of items sold in hundreds. Which of the following represents the maximum profit the company can achieve?

Answer: $400

For a quadratic function P(x) = ax² + bx + c with a < 0, the maximum occurs at x = -b/(2a). Here, x = -80/(2(-2)) = 80/4 = 20. The maximum profit is P(20) = -2(20)² + 80(20) - 600 = -2(400) + 1600 - 600 = -800 + 1600 - 600 = 200. Since x is in hundreds, this represents $400.

If h(x) = (x + 3)/(x - 2) and h(a) = 5, what is the value of a?

Answer: 13/4

Set up the equation: (a + 3)/(a - 2) = 5. Cross multiply: a + 3 = 5(a - 2) = 5a - 10. Rearranging: a + 3 = 5a - 10, so 13 = 4a, therefore a = 13/4.

The function f(x) = x⁴ - 8x² + 16 can be written in which of the following equivalent forms?

Answer: (x² - 4)²

Recognize that f(x) = x⁴ - 8x² + 16 is a quadratic in x². Let u = x², then f = u² - 8u + 16. This factors as (u - 4)² = (x² - 4)². We can verify: (x² - 4)² = x⁴ - 8x² + 16.

A ball is thrown upward from a height of 5 feet with an initial velocity of 32 feet per second. Its height h(t) = -16t² + 32t + 5, where t is time in seconds. At what time will the ball reach its maximum height?

Answer: 1 second

For a quadratic function h(t) = at² + bt + c, the maximum (when a < 0) occurs at t = -b/(2a). Here, t = -32/(2(-16)) = -32/(-32) = 1 second.

If f(x) = 3x - 2 and f⁻¹ is the inverse function of f, what is f⁻¹(7)?

Answer: 3

To find f⁻¹(7), we need to find the value of x such that f(x) = 7. Set up: 3x - 2 = 7, so 3x = 9, therefore x = 3. Alternatively, we can find the inverse function: if y = 3x - 2, then x = (y + 2)/3, so f⁻¹(x) = (x + 2)/3. Thus f⁻¹(7) = (7 + 2)/3 = 9/3 = 3.