Functions — 700-800

Functions: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

At the 700-800 level, function questions are almost never "evaluate $f(3)$." They are chains: two or three rules stacked end to end, with the real difficulty living in which link of the chain the question actually asks about.

Archetype 1: The Rate Chain (Composition in Disguise)

A stem gives you two or three rates in different units and never uses the word "composition."

A press prints $45$ pages per minute. The cost of a job of $p$ pages is $f(p) = 0.08p + 12$, where \12$is a per-job setup fee. A job cost$$48$. How many minutes did the press run?

The chain is minutes $\rightarrow$ pages $\rightarrow$ dollars. Going backward: $0.08p + 12 = 48$ gives $p = 450$ pages, then $450 \div 45 = 10$ minutes.

Both intermediates are planted. $450$ (pages) is a choice. So is $13\frac{1}{3}$ — what you get if you forget the setup fee. So is $0.8$ — what you get if you divide the leftover \36$by the *page* rate$45$instead of the *cost-per-minute* rate$3.6$.

The discipline: write the chain with units on one line before touching numbers — minutes ($\times 45$) $\rightarrow$ pages ($\times 0.08$, then $+12$) $\rightarrow$ dollars. Then read the last sentence of the stem and stop at the link it names — not at the link where the algebra felt finished.

Archetype 2: Working Backward Through a Table

$f(1) = 6$, $f(2) = 3$, $f(3) = 7$, $f(4) = 1$, $f(5) = 4$. If $g(x) = 2x - 1$ and $f(g(x)) = 4$, what is $x$?

Work outside in. $f(\text{something}) = 4$ forces that something to be $5$, because $f(5) = 4$. So $g(x) = 5$, and $2x - 1 = 5$ gives $x = 3$.

The three planted errors, all present as choices:

• $5$ — you found $g(x)$ and stopped. The single most common miss.
• $9$ — you found that the $f$-input must be $5$, then applied $g$ forward ($g(5) = 9$) instead of undoing it.
• $2.5$ — you ignored $f$ entirely and solved $2x - 1 = 4$.

The same structure appears with an inner transformation: if $g(x) = f(2x) + 3$ and $g(x) = 10$, then $f(2x) = 7$, the table gives $2x = 3$, and the answer is $x = 1.5$ — with $3$ sitting right there as a choice.

Archetype 3: Transformations — Inside Changes Input, Outside Changes Output

Every hard transformation item tests the same two rules:

• Inside the parentheses affects $x$, and does the opposite of what it looks like. $f(x - 3)$ shifts right $3$. $f(2x)$ compresses horizontally by a factor of $2$ (points move toward the $y$-axis, so their $x$-coordinates get halved).
• Outside affects $y$, and does exactly what it looks like. $f(x) + 4$ shifts up $4$; $-f(x)$ reflects across the $x$-axis; $1.2f(x)$ makes every height $20\%$ larger.

The order trap. "Translated $3$ right and $2$ down, then reflected across the $x$-axis" is not $-f(x - 3) - 2$. The reflection negates everything already there: $g(x) = -\big[f(x - 3) - 2\big] = -f(x - 3) + 2$.

If $f(1) = -4$, then $g(4) = -(-4) + 2 = 6$. The choice $-6$ is "reflection forgotten," and the choice $2$ is "reflected the $f$ term but left the shift alone."

The word-problem version. "A second plant was sown $5$ days later and is $20\%$ taller at every corresponding age." Later planting $\rightarrow$ it is younger at time $t$ $\rightarrow$ inside becomes $t - 5$. Taller $\rightarrow$ output scaling $\rightarrow$ $1.2$ outside. Answer: $1.2h(t - 5)$. The wrong answers put the $20\%$ inside ($h(1.2t - 5)$) or turn the $5$-day delay into a $5$-centimeter drop ($1.2h(t) - 5$).

Archetype 4: Composition Order — Match the Units

A stem gives two rules and asks for one expression. Suppose $h(B)$ gives the number of hours of shop time a budget of $B$ dollars buys, and $c(t)$ gives the number of bicycles a crew can assemble in $t$ hours. The chain is dollars $\rightarrow$ (apply $h$) $\rightarrow$ hours $\rightarrow$ (apply $c$) $\rightarrow$ bicycles, so the number of bicycles a budget of $B$ dollars produces is $c(h(B))$.

The inner function is the one whose input matches what you HAVE (dollars); the outer function is the one whose output matches what you WANT (bicycles). The choice $h(c(B))$ has the right functions in the wrong order — it feeds a dollar amount into $c$, which expects hours. The choice $h(B)$ stops one link early and produces hours, not bicycles. The choice $h(B) \cdot c(B)$ multiplies two outputs instead of chaining them.

When a rule is defined by working backward, evaluate the inside first. If $g(x)$ is the value of $t$ for which $f(t) = x - 6$, and $f(3) = 11$, then $g(17)$ is the value of $t$ for which $f(t) = 11$, which is $3$. The choice $11$ is the inner value. The other planted route solves $f(t) = 17$ first and then subtracts $6$ from that $t$, which moves the shift outside.

Functions: Traps & Speed

Part 2 of 3 — The Four Species of Wrong Answer

Hard function items are not written by adding hard arithmetic. They are written by taking a correct multi-step solution and turning each intermediate step into an answer choice. Learn the four species and you can often eliminate three options before finishing the algebra.

Species 1: The Intermediate Value (the big one)

Roughly two of every three hard function items plant the value you compute one step before the end.

The question asks for	The planted intermediate
minutes	the number of pages / gallons / bottles
cases shipped	the number of pallets
milligrams per dose	the milligrams per day
the amount over budget	the total cost
elapsed time between two events	the second event's time
seconds	the radius, then the area

The habit that beats it: before you click, say the units of your number out loud and compare them with the last four words of the stem. "$20$ pallets" versus "how many CASES" is a mismatch you can catch in two seconds.

A special case worth its own line: the "additional / more than / exceeds" ask. When a stem ends with "by how much does it exceed," "how many MORE," or "how many additional," you have one subtraction left after the number that feels final.

Species 2: The Dropped Fixed Fee

Any model of the form $f(x) = mx + b$ in context is an invitation. Divide the whole output by $m$ and you get a clean-looking wrong answer every time.

• Correct: \83.36 = 0.992m + 4 \Rightarrow 0.992m = 79.36 \Rightarrow m = 80$.
• Planted: $83.36 \div 0.992 = 84.0$.

Note how reasonable $84$ looks. Fixed fees must be stripped before any division.

Species 3: The Wrong-Rate Division

You correctly isolate a quantity, then divide by a rate from the wrong link of the chain: dollars divided by miles-per-gallon, leftover cost divided by trays-per-hour, budget divided by liters-per-bottle instead of dollars-per-bottle. Every one of these produces a plausible number in impossible units. Cancel units in the division, not just the numbers.

Species 4: Direction Errors in Chains and Transformations

Applying $g$ forward when the situation calls for working backward from its output; shifting left when the story says the second thing started later; stretching when the rule says $f(2x)$; nesting two functions in the wrong order; multiplying by an exchange rate when the conversion required dividing.

The size check catches almost all of these. If a euro is worth more than a dollar, the dollar figure must be bigger. If the second plant was sown later, it must be shorter at time $t$ (before its $20\%$ advantage is applied). If a compression pulls points toward the $y$-axis, the $x$-coordinate must shrink.

Speed Techniques

1. Collapse the rate chain into one constant. Do not compute step by step through a three-rate stem. Multiply the rates together first and watch for the arithmetic gift the writers hide there.

$1.5$ scoops per minute, $0.8$ kilogram per scoop, \1.25$ per kilogram.

Note $0.8 \times 1.25 = 1$ exactly, so the whole chain collapses to \1.50$per minute and the inequality$1.5t \le 60$ is a mental calculation. Hard items are built around such collapses; if the numbers look ugly, you probably multiplied in the wrong order.

2. Outside-in for compositions, inside-out for evaluations. If the equation is $f(g(x)) = k$, start at the outer function and read the table backward. If the expression is $g(f(3))$, start inside. Choosing the wrong direction is what produces the swapped-intermediate distractors.

3. On "which expression" items, track units, not algebra. For $g(f(1250)) - g(f(900))$: $f$ outputs gallons, $g$ turns gallons into dollars, and a difference of two dollar amounts is dollars. That single pass eliminates every choice that ends in gallons or in a per-unit rate.

Then apply the proportionality check: unless the stem says $f$ and $g$ are proportional, $g(f(1250)) - g(f(900))$ is not $g(f(350))$. A setup fee or bulk discount breaks it. "The cost of the paint for $350$ square feet" is the most attractive wrong answer on the whole test for this archetype.

4. Discrete quantities round UP, boundary values are excluded. Spools, buses, and shifts come in whole units: $4.85$ spools means buy $5$. Both $4$ (rounded down) and $4.85 \times \text{price}$ (fractional spool) will be options.

And when a stem says fewer than or more than, the boundary is not the answer. If $\frac{300}{4 + 2d} < 25$ gives $d > 4$, then day $4$ — where the job takes exactly $25$ hours — is planted, and the answer is day $5$.

5. Budget your time by the number of links. A two-link chain should take $60$ seconds; a three-link chain with a unit conversion, about $90$. If you are past two minutes, you have almost certainly mis-ordered the chain — rewrite it with units and restart rather than pushing forward.

Functions: Timed Drill

Part 3 of 3 — Four Questions at Full Difficulty

Work at about 90 seconds per question — the realistic budget for a hard Module 2 function item. For each one, run the same three-beat routine:

1. Ten seconds: name the archetype (rate chain, table reversal, transformation, composition order) and write the chain with units.
2. Sixty seconds: compute, keeping every intermediate labeled with its units.
3. Twenty seconds: re-read the final sentence and confirm your number wears the units it asks for. If your value appears among the choices in the wrong units, that is confirmation you built the right chain — and a warning that the trap is one step behind you.

Do not look at the answer choices before step 2. On these items the choices are engineered to make each intermediate feel like a destination.