AMC 10 · 2002 · #19
Grade 9 algebra
Pick an answer.
There is no formula for f, only a picture, and the equation wraps f inside itself. Tool #11 (Work Backwards) handles the wrapping: the outer f must land on 6, so ask first which inputs t satisfy f(t)=6, and only then hunt for the x that produce those t. Tool #4 (Introduce a Variable) makes that concrete by naming t=f(x). Tool #1 (Draw a Diagram) is used in reverse — the diagram is given, so read the four corner points off it and convert each segment into a linear equation with its own interval. Tool #7 (Identify Subproblems) then splits the work into two ordinary equations, f(x)=-2 and f(x)=1, each solved four times, once per piece. Tool #2 (Make a Systematic List) collects the surviving roots at the end so none is counted twice and none is dropped. Solving on pieces rather than eyeballing crossings is what keeps the count honest.
Turn the graph into four lines
Corner to corner the slopes give four formulas, each valid only on its own stretch of [-7,5].
Two corner points pin a straight piece completely, so a picture of segments is really a short list of equations.
8.F.B.4Draw A DiagramPeel the outer f first
Peel the outer f: f(t)=6 holds exactly at the two peaks t=-2 and t=1.
Before chasing x, find out what the outer f needs to be fed; everything else is a search for those two numbers.
8.EE.C.7Work BackwardsSplit into two plain equations
So the nested equation splits into the two plain ones f(x)=-2 or f(x)=1.
A composition is two rules stacked; strip the outer rule and what remains is an equation you already know how to solve.
9.F-IF.A.2Introduce A VariableSolve f(x)=-2 on each piece
Solving f(x)=-2 on each piece and keeping only in-range answers leaves 2 solutions.
A formula that is only valid on part of the line can hand back an answer from outside that part, so every root has to prove it belongs.
8.EE.C.7Identify SubproblemsSolve f(x)=1 on each piece
Solving f(x)=1 the same way, all four pieces survive, giving 4 solutions.
The target 1 sits low enough to be crossed by every piece, while -2 is below two of them entirely.
8.EE.C.7Identify SubproblemsCombine the two lists and count
A function cannot output both targets at once, so the counts add: 6, choice (D).
Cases built from different outputs of one function can never collide, so their counts add with nothing subtracted.
Roots coming from the two different inner values can never coincide, so the two counts simply add.
▸ Why?
A single input has one output, so no x can make the inner function equal both targets at once.
▸ Why?
The two families of roots cover every solution and never overlap, so adding their sizes counts each solution once.
When a function is wrapped inside itself, solve the outside first: find every value the outer f needs to be given, then go hunting for the x that produce them.
- Turn the graph into four lines
- Peel the outer f first
- Split into two plain equations
- Solve f(x)=-2 on each piece
- Solve f(x)=1 on each piece
- Combine the two lists and count