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For all numbers x, two functions are defined by f(x) = 3x − 4 and g(x) = 2x + 5. A planner takes a number n, first applies g to it, and then applies f to that result. The final output equals 8 more than 5 times n. What is n?
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Correct answer
B
Translate the prose into a composition in the order stated: g is applied first, then f, so the output is f(g(n)). Build it from the inside out. The inner function gives g(n) = 2n + 5. Substitute that whole expression into f, keeping every constant: f(g(n)) = 3(2n + 5) − 4 = 6n + 15 − 4 = 6n + 11.
The phrase "8 more than 5 times n" translates to 5n + 8. Set the two equal: 6n + 11 = 5n + 8. Subtract 5n from both sides to get n + 11 = 8, then subtract 11 to get n = −3. So the answer is (B).
Method check by plugging back: g(−3) = 2(−3) + 5 = −1, then f(−1) = 3(−1) − 4 = −7, and 5(−3) + 8 = −7, which matches.
The wrong choices each come from one specific slip. (A) −7 drops the −4 in f during substitution, so the −4 never enters the equation. (C) 3 sets up the right equation 6n + 11 = 5n + 8 but reverses the final subtraction, computing 11 − 8 instead of 8 − 11. (D) 11 reverses the order of composition, applying f before g to get g(f(n)) = 6n − 3 instead of f(g(n)). (E) 12 drops the +5 inside g, so f sees only 2n rather than 2n + 5. The defensible path requires composing in the stated order and carrying both constants through the substitution.
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