Math: Advanced Functions Flashcards

The function f is defined by f(x) = x^2 - 8x + 15. The graph of y = f(x) in the xy-plane is a parabola. Which of the following is the best interpretation of the vertex of the parabola?

Answer: The minimum value of f(x) is -1, which occurs at x = 4.

To find the vertex of the parabola defined by the quadratic function f(x) = ax^2 + bx + c, we can use the formula x = -b/(2a) to find the x-coordinate. In this case, a = 1 and b = -8. So, x = -(-8)/(2*1) = 8/2 = 4. To find the y-coordinate, which represents the minimum value of the function since the parabola opens upwards (a > 0), we substitute x = 4 back into the function: f(4) = (4)^2 - 8(4) + 15 = 16 - 32 + 15 = -1. Therefore, the vertex is at (4, -1), and the minimum value of the function is -1.

A biologist models the population of a certain species of bacteria with the function P(t) = 100(2)^(t/3), where P(t) is the population after t hours. Which of the following equations best models the amount of time, t, in hours, it will take for the population to reach 1,600?

Answer: 1600 = 100(2)^(t/3)

The question asks for the equation that models the time 't' it takes for the population P(t) to reach 1,600. The given function is P(t) = 100(2)^(t/3). To find the time when the population is 1,600, we need to set P(t) equal to 1,600. This results in the equation 1600 = 100(2)^(t/3).

If f(x) = 3x + 7 and g(x) = x^2 - 1, what is the value of f(g(3))?

Answer: 31

This is a composition of functions problem. First, we need to evaluate the inner function, g(3). g(3) = (3)^2 - 1 = 9 - 1 = 8. Now we take this result and use it as the input for the outer function, f(x). So we evaluate f(8): f(8) = 3(8) + 7 = 24 + 7 = 31.

The function h is defined by h(x) = (x-5)^3 + 2. For which of the following values of x is h(x) = 10?

Answer: 7

We are asked to find the value of x for which h(x) = 10. We set up the equation: 10 = (x-5)^3 + 2. First, subtract 2 from both sides: 8 = (x-5)^3. Next, take the cube root of both sides: ∛8 = x-5. Since ∛8 = 2, we have 2 = x-5. Finally, add 5 to both sides to solve for x: x = 7.

Which of the following expressions is equivalent to (x^2 - 9)/(x+3) for x ≠ -3?

Answer: x - 3

The expression x^2 - 9 is a difference of squares, which can be factored into (x-3)(x+3). The full expression is therefore ((x-3)(x+3))/(x+3). Since x ≠ -3, the (x+3) term in the numerator and denominator can be cancelled out, leaving x - 3.

The graph of the function f(x) = a(x+1)(x-3) is a parabola that passes through the point (1, -8). What is the value of a?

Answer: 2

The problem states that the point (1, -8) is on the graph of the function. This means that when x = 1, f(x) = -8. We can substitute these values into the function's equation to solve for 'a'. -8 = a(1+1)(1-3). This simplifies to -8 = a(2)(-2), which is -8 = -4a. Dividing both sides by -4 gives a = 2.