AP PreCalculus Practice Quiz: Composition of Functions

Correct Answer: B

To evaluate (f ◦ g)(4), first calculate g(4), which is 4^2 = 16. Then, use this output as the input for f: f(16) = 3(16) + 2 = 48 + 2 = 50. This follows the principle of evaluating the composition of two functions for a given value.

Correct Answer: A

To construct the representation of f(g(x)), we substitute the expression for g(x) into every instance of x in f(x). So, f(g(x)) = f(2x) = 5(2x) - 1 = 10x - 1.

Correct Answer: C

The definition of a composite function f ◦ g states that it maps a set of input values to a set of output values such that the output values of g are used as input values of f.

Correct Answer: A

(f ◦ g)(x) = f(g(x)) = f(4x) = 4x + 3. (g ◦ f)(x) = g(f(x)) = g(x+3) = 4(x+3) = 4x + 12. Since 4x + 3 ≠ 4x + 12, this shows that f ◦ g and g ◦ f are different functions.

Correct Answer: B

To decompose h(x), we identify an 'inner' function g(x) and an 'outer' function f(x). Here, the inner function is g(x) = 2x - 5. The outer function takes this input and cubes it, so f(x) = x^3. Then f(g(x)) = f(2x - 5) = (2x - 5)^3.

Correct Answer: C

When the identity function f(x) = x is composed with any function g, the resulting composite function is the same as g. Therefore, (f ◦ g)(x) = f(g(x)) = g(x) = √x + 7.

Correct Answer: B

To find f(g(x)), we substitute the entire expression for g(x) into x in the function f(x). This gives f(g(x)) = f(x+1) = 1/((x+1) - 2) = 1/(x-1).

Correct Answer: A

Functions can often be decomposed into less complicated functions. For h(x) = √(x^2 + 9), the 'inner' operation is squaring x and adding 9, so g(x) = x^2 + 9. The 'outer' operation is taking the square root, so f(x) = √x. This gives f(g(x)) = f(x^2 + 9) = √(x^2 + 9).

Correct Answer: A

This is a composition of three functions. We evaluate from the inside out. First, h(-3) = (-3)^2 = 9. Next, we use this result in f: f(9) = 2(9) = 18. Finally, we use this result in g: g(18) = 18 - 5 = 13.

Correct Answer: D

The expression (g ◦ f)(x) means g(f(x)). If g(f(x)) = g(x), it implies that the input to g, which is f(x), must be equal to x. Therefore, f(x) must be the identity function, f(x) = x.

Correct Answer: A

To evaluate (g ◦ f)(1), we first find the value of the inner function, f(1). f(1) = 1^2 - 4 = -3. Then, we use this output as the input for the outer function, g: g(-3) = 3(-3) = -9.

Correct Answer: A

To construct f(g(x)), we substitute g(x) = 3x for every instance of x in f(x). This gives f(3x) = 2(3x) / ((3x)-1) = 6x / (3x-1).

Correct Answer: B

We need to find an inner function g(x) and an outer function f(x). The expression x^2+1 is inside the parentheses, making it a good candidate for the inner function, so g(x) = x^2+1. The outer function then takes this input, raises it to the 5th power, and multiplies by 4. Thus, f(x) = 4x^5. Checking: f(g(x)) = f(x^2+1) = 4(x^2+1)^5.

Correct Answer: B

The provided content explicitly states that the composition of functions is not commutative; that is, f ◦ g and g ◦ f are typically different functions. The order in which the functions are composed matters.

Correct Answer: A

To find (g ◦ f)(x), which is g(f(x)), we substitute the expression for f(x) into the variable x in g(x). So, g(f(x)) = g(x^2 + 2x) = (x^2 + 2x) - 1 = x^2 + 2x - 1.

Correct Answer: C

The content states: 'When analytic representations of the functions f and g are available, an analytic representation of f(g(x)) can be constructed by substituting g(x) for every instance of x in f.' This is the direct procedure for finding the composite function's formula.