AMC 10 · 2007 · #9

Grade 9 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

function-evaluationfunction-compositionlinear-equations-one-var work-backwards ↑ Prerequisites: function-evaluationlinear-equations-one-var

📏 Short solution 💡 2 insights

Problem

A rule is described only through what it does to a shifted input. Find what it returns for one particular value.

Pick an answer.

(A)

(B)

(C)

(D)

111

(E)

211

How to solve

Strategy Work Backwards

The rule is handed to us at the input 3x-1, but the question asks about the input 5. So do not push forward from a chosen x; work backwards from the target input (Tool #11) and ask which x makes the fed-in number equal 5. That is a one-line linear equation, and the whole problem rests on the fact that it has exactly one solution: 3x-1 has slope 3 ≠ 0, so different x values land on different inputs and f(5) is pinned down without ambiguity. To check the value by a route with a different shape, rename the input (Tool #4) as t=3x-1 and rewrite the entire rule as a formula in t (Tool #15). That formula also settles a question the substitution route quietly assumes — that a function with this property exists at all.

1STEP 1

Notice what f is actually fed

The letter is not the input, so substituting it answers the wrong question.

x=5: f(3 · 5-1)=f(14)=5²+5+1=31 (this is f(14), not f(5))

2STEP 2

Work backwards to the right x

Working backwards finds the letter that makes the input right.

3x-1=5 ⟺ 3x=6 ⟺ x=2

3STEP 3

Evaluate the right-hand side at x=2

Evaluating there gives 7.

f(5)=f(3 · 2-1)=2²+2+1=7

4STEP 4

Confirm by building f itself

Building the rule outright confirms 7, choice (A).

f(t)=(t²+5t+13)/9, f(5)=(25+25+13)/9=63/9=7

Answer

Test the built formula at a value we never used: x=1 makes the original identity say f(2)=1+1+1=3, and the formula gives (2²+5 · 2+13)/9=27/9=3. They agree. The four wrong choices are all the same mistake in different clothes — each is x²+x+1 evaluated at some x whose matching input is not 5: 13 comes from x=3 and is f(8), 31 comes from x=5 and is f(14), 111 comes from x=10 and is f(29), and 211 comes from x=14 and is f(41). In particular 31 is the trap for anyone who substitutes 5 where x stands instead of where the input stands. It is also reasonable that the true value is the smallest choice: hitting the input 5 pulls x back to 2, and x²+x+1 is small when x is small.

💡Key takeaway

When a rule is written as f(something), first ask what number is sitting inside the parentheses, then work backwards to the x that puts your target number there.

Notice what f is actually fed
Work backwards to the right x
Evaluate the right-hand side at x=2
Confirm by building f itself

Notice what f is actually fed

Work backwards to the right x

Evaluate the right-hand side at x=2

Confirm by building f itself

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