Write a function for a relationship between two quantities as an explicit expression, a recursive process or steps for calculation, then build new functions by combining and composing standard types

Learning goal: Write a function that describes a relationship between two quantities. Starting from a context, determine an explicit expression, a recursive process, or steps for calculation that produce the output from the input. Then build more complicated functions from standard function types, linear, quadratic and exponential, by combining them with arithmetic operations and by composing one function with another, and relate each piece of the new function to the situation it models.

Scope note. Part c of F.BF.1 (composing functions) is marked (+) in Ohio's Learning Standards for mathematics, which Ohio uses to indicate a standard "for students who are planning on taking advanced courses". It is taught here in full, as extension content for students on that path.

What it means to write a function from context. A situation describes how one quantity depends on another: the cost of a gym membership depends on the number of months, the height of a ball depends on the time since it was thrown, the balance of an account depends on the number of years it has earned interest. To write a function is to state the rule that turns the input into the output, and to say what the input and output are, with units. There are three ways to state the rule, and a good modeler chooses the one the situation hands over most naturally.

An explicit expression. An explicit expression computes the output directly from the input in one formula. Suppose a hypothetical gym charges a joining fee of 30 dollars and then 25 dollars for each month of membership. The total cost of m months is C(m) = 25m + 30 dollars. The expression is explicit because any month count goes straight into the formula: C(12) = 25 · 12 + 30 = 330 dollars, with no need to know the cost of the eleven months before. The 25 is the rate, dollars per month, and the 30 is the starting value, the cost before any month has passed.

A recursive process. A recursive process states the first output and then tells how each output is produced from the one before it. The same gym cost is described recursively by C(1) = 55 and C(m) = C(m − 1) + 25 for m ≥ 2. To find C(3) recursively, compute C(2) = 55 + 25 = 80 and then C(3) = 80 + 25 = 105, which agrees with the explicit 25 · 3 + 30 = 105. A recursive process is the natural choice when the situation itself is described step by step. An account that starts with 500 dollars and earns 3 percent interest each year, compounded annually, is described by A(0) = 500 and A(n) = 1.03 · A(n − 1), because the situation says what happens each year: multiply by 1.03. The explicit expression for the same account is A(n) = 500(1.03)ⁿ. Both give A(2) = 530.45 dollars: recursively, 500 · 1.03 = 515.00 and 515.00 · 1.03 = 530.45; explicitly, 500 · 1.0609 = 530.45. After five years the explicit expression gives A(5) = 500(1.03)⁵ ≈ 579.64 dollars. The recursive process is easy to state and easy to compute one step at a time; the explicit expression is what you need to jump to year 40 without computing the 39 years before.

Steps for calculation. Sometimes the cleanest description is an ordered list of steps. A farmer has 100 meters of fencing and will enclose a rectangular pen against a straight wall, so only three sides need fence. Let x be the length, in meters, of each of the two sides perpendicular to the wall. The steps for calculating the area are: double x; subtract that from 100 to get the length of the side parallel to the wall; multiply that length by x. Written as an explicit expression the steps become A(x) = x(100 − 2x) = 100x − 2x², a quadratic function. The steps are the reasoning; the expression is its compressed record. Check: x = 20 gives a parallel side of 60 and an area of 1,200 square meters; x = 25 gives a parallel side of 50 and an area of 1,250 square meters, the maximum, at the vertex of the parabola.

Which functions each course writes from context. Ohio sets a focus for this skill by course. In Algebra 1 and Math 1 the focus is on linear and exponential functions, like the gym cost and the account above. In Algebra 1 and Math 2 it is on situations that exhibit quadratic or exponential relationships, like the fenced pen and the account. The practice questions ask for all three: a linear cost, an exponential balance and a quadratic area.

Combining standard function types with arithmetic operations. New functions are built from standard function types by adding, subtracting, multiplying and dividing them, and each operation has a meaning in the model. Profit is revenue minus cost: if a maker sells x lamps at 40 dollars each, revenue is R(x) = 40x, and if the costs are 1,500 dollars of fixed costs plus 15 dollars per lamp, cost is C(x) = 1500 + 15x, so profit is P(x) = R(x) − C(x) = 40x − 1500 − 15x = 25x − 1500. Subtracting two linear functions gave a linear function whose slope 25 is the profit per lamp and whose zero, x = 60, is the break-even point, since P(60) = 1500 − 1500 = 0. Multiplying two non-constant linear functions produces a quadratic: if the price a stall can charge falls as it offers more, say price p(x) = 50 − 0.5x dollars when x items are offered, then revenue is x · p(x) = 50x − 0.5x², and x = 50 gives revenue 50 · 25 = 1,250 dollars while x = 100 gives 100 · 0 = 0.

The example every student should know is the temperature of a cooling body. A cup of coffee at 80 °C sits in a room at 20 °C. What cools is the difference between the coffee and the room, which starts at 60 degrees and, in a simple model, shrinks by the same factor each minute, say to 0.8 of itself. The difference is a decaying exponential, 60(0.8)ᵗ, and the coffee's temperature is that difference added to the constant room temperature: T(t) = 20 + 60(0.8)ᵗ degrees Celsius after t minutes. Adding a constant function to a decaying exponential is exactly what the model demands, and each piece relates to the situation. The constant 20 is the room temperature, the value the coffee approaches and never passes. The 60 is the initial difference, so T(0) = 20 + 60 = 80. The factor 0.8 says the gap closes by 20 percent per minute. After 5 minutes, T(5) = 20 + 60(0.8)⁵ = 20 + 60 · 0.32768 ≈ 39.66 °C; after 10 minutes, T(10) = 20 + 60 · 0.10737 ≈ 26.44 °C. A student who wrote 80(0.8)ᵗ alone would have the coffee cooling toward 0 °C in a 20 degree room, which is the misconception the constant function prevents.

Composing functions. Composition feeds the output of one function into another. If T(y) is the temperature in the atmosphere as a function of height y, and h(t) is the height of a weather balloon as a function of time t, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Suppose, in a simplified model, that the temperature is T(y) = 15 − 6.5y degrees Celsius at height y kilometers, and that the balloon rises at a steady rate so that h(t) = 0.3t kilometers after t minutes. Then T(h(t)) = 15 − 6.5(0.3t) = 15 − 1.95t. After 10 minutes the balloon is at h(10) = 3 kilometers, where the temperature is T(3) = 15 − 19.5 = −4.5 °C, and the composed function gives the same −4.5 directly. The composition only makes sense in one direction: h(T(y)) would feed a temperature into a function that expects a time, which is meaningless. Order matters even when both orders make sense. A store takes 20 dollars off a coat, d(p) = p − 20, and then tax adds 7 percent, t(p) = 1.07p. Discount first, then tax: t(d(p)) = 1.07(p − 20) = 1.07p − 21.40. Tax first, then discount: d(t(p)) = 1.07p − 20. For a 100 dollar coat these are 85.60 and 87.00 dollars, and the shopper should hope the store composes in the first order.

Worked explicit and recursive descriptions: A hypothetical phone plan costs 40 dollars the first month and 32 dollars each month after. Write the total paid after m months explicitly and recursively. Recursively, P(1) = 40 and P(m) = P(m − 1) + 32 for m ≥ 2, so P(2) = 72 and P(3) = 104. Explicitly, m months contain m − 1 months at 32 dollars after the first, so P(m) = 40 + 32(m − 1) = 32m + 8. Check P(3) = 96 + 8 = 104, agreeing with the recursion. The explicit form answers P(24) = 768 + 8 = 776 dollars at once.

Worked combination of standard types: A pond is stocked with 200 fish, and the population is expected to double every 3 years, but a fixed group of 50 fish is kept in a separate tank that does not breed. Write a function for the total number of fish after t years. The breeding population is exponential, 200 · 2^(t/3), and the tank holds a constant 50, so the total is F(t) = 50 + 200 · 2^(t/3). At t = 3, F(3) = 50 + 400 = 450; at t = 6, F(6) = 50 + 800 = 850. The constant relates to the tank, the 200 to the initial stocked group, and the base 2 with exponent t/3 to doubling every 3 years.

Worked composition: An oil slick on still water spreads as a circle whose radius grows at 2 meters per minute, so r(t) = 2t meters after t minutes. The area of a circle of radius r is A(r) = πr². The area as a function of time is the composition A(r(t)) = π(2t)² = 4πt² square meters. At t = 5 minutes the radius is r(5) = 10 meters and the area is A(10) = 100π ≈ 314.16 square meters; the composed formula gives 4π · 25 = 100π, the same value. The reverse order, r(A(r)), would feed an area into a function expecting a time and is not meaningful here.

Representation task: For the cooling coffee T(t) = 20 + 60(0.8)ᵗ, make a table for t = 0, 1, 2, 3, 4, 5 and 10 minutes, showing three columns: the decaying difference 60(0.8)ᵗ, the constant 20, and their sum. Plot the sum and draw the horizontal line y = 20. Under the graph write one sentence for each of the three numbers 20, 60 and 0.8 saying what it means for the coffee, and one sentence explaining what would be wrong with the model 80(0.8)ᵗ.

Independent performance: Choose a situation of your own that involves a fixed start and a repeated step (a fee plus a rate, or a starting amount multiplied by a factor each period). Write the function three ways: as steps for calculation, as a recursive process with its first value, and as an explicit expression, and show that all three give the same output for one input beyond the second. Then build two more functions: one that combines two standard function types with an arithmetic operation (a sum, difference or product) and one that composes two functions, in the style of temperature as a function of height composed with height as a function of time. For each built function, state what every constant means in the situation, compute one output, and check it by computing the pieces separately. Success criteria: the recursive process states a first value and a rule for each later value; the explicit expression agrees with the recursion at the checked input; each combined or composed function is written in simplified form with units; and the order of composition is justified by which quantity is the input of which function.