SAT Math Multiple Choice Question 537: Answer and Explanation
12. If f(x) = x – 1, g(x) = x3, and x ≤ 0, which of the following could not be in the range of f(g(x))?
- A. –27
- B. –3
- C. –1
- D. 1
Correct Answer: D
Category: Passport to Advanced Math / Functions
Strategic Advice: Sometimes, a question requires thought rather than brute force. Here, you need to understand that when dealing with compositions, the range of the inner function becomes the domain of the outer function, which in turn produces the range of the composition.
Getting to the Answer: In the composition f(g(x)), the function g(x) = x3 is the inner function. Because the question states that x is either zero or a negative number (x ≤ 0), every value of x, when substituted into this function, will result in zero or a negative number (because a negative number raised to an odd power is always negative). This means that the largest possible range value for g(x) is 0, and consequently that the largest possible domain value for f(x) is also 0. Substituting 0 for x in f(x) results in –1, which is the largest possible range value for the composition. Because 1 > –1, it is not in the range of f(g(x)), so (D) is correct.