Practice Maths

L47 — Applying Functions to Real Problems

Key Terms

Break-even point
The output level where Revenue = Cost, giving zero profit; found by solving R(n) = C(n).
Revenue function R(n)
R(n) = price × quantity — the total income from selling n units.
Cost function C(n)
C(n) = (variable cost per unit) × n + fixed costs — total cost to produce n units.
Profit function P(n)
P(n) = R(n) − C(n); positive means profit, negative means loss.
Contextual domain
The subset of the mathematical domain that makes physical sense (e.g. n ≥ 0, n ∈ ℤ for "number of items").
Contextual range
The set of output values achievable within the contextual domain — often more restricted than the mathematical range.

Domain and range in context

FunctionContextNatural domainRange
C(h) = 45h + 60Plumber's cost for h hoursh ≥ 0C ≥ 60
h(t) = −5t² + 20tBall height (m) at time t0 ≤ t ≤ 40 ≤ h ≤ 20
P(n) = 2000 × 1.05nInvestment balance after n yearsn ≥ 0, n ∈ ℤP ≥ 2000
Hot Tip: When the domain is restricted in context, always state the restriction explicitly (e.g. n ≥ 0, n ∈ ℤ). An answer of n = 37.5 items makes no sense — round up to n = 38 and verify it satisfies all conditions.

Worked Example — Break-Even Analysis

Situation: A t-shirt business has fixed costs of $800 and produces each shirt for $12. Each shirt sells for $20.

Step 1 — Write cost and revenue functions.
Let n = number of shirts. Cost: C(n) = 12n + 800. Revenue: R(n) = 20n.

Step 2 — Find break-even point.
Set R = C: 20n = 12n + 800 ⇒ 8n = 800 ⇒ n = 100 shirts.

Step 3 — Write profit function.
Profit P(n) = R(n) − C(n) = 20n − (12n + 800) = 8n − 800.

Step 4 — Interpret.
For n > 100: profit. For n < 100: loss. Each extra shirt above 100 adds $8 profit.

Functions Describe Real-World Processes

Every time a quantity changes in response to another quantity, there is potentially a function hiding in the situation. The skill is identifying the input (independent variable), the output (dependent variable), and the rule connecting them.

Domain Restrictions in Context

Mathematically, f(x) = √x is defined for x ≥ 0. But in context, there may be additional restrictions. If x is the number of workers on a site and the minimum is 2, then the domain is x ≥ 2. Always ask: what values of x are physically possible?

Worked Example 2 — Cost, Revenue and Profit

A cake shop sells slices for $6 each. Fixed costs are $150/day. Variable cost is $2/slice. Find the profit function and the minimum number of slices needed to make a profit.

Solution

Let n = slices sold per day.
Revenue: R(n) = 6n.
Cost: C(n) = 2n + 150.
Profit: P(n) = R − C = 6n − 2n − 150 = 4n − 150.

For profit > 0: 4n − 150 > 0 ⇒ n > 37.5. So minimum 38 slices per day.

Domain in context: n ≥ 0, n ∈ ℤ (whole slices only).

Worked Example 3 — Composite Functions in Context

Water flows into a tank at r(t) = 3t litres per minute (accelerating). The cost of water is c(v) = 0.4v dollars for v litres.

Solution

After t minutes, volume = r(t) = 3t litres.
Cost: c(r(t)) = 0.4 × 3t = 1.2t dollars.

After 10 minutes: cost = $12. The composition c(r(t)) gives cost as a direct function of time.

Worked Example 4 — Inverse Function in Context

Temperature conversion: F(c) = 1.8c + 32 converts Celsius to Fahrenheit.

Solution

Inverse: swap C and F. c = (F − 32)/1.8. So F−1(F) = (F − 32)/1.8.

At F = 98.6°F (body temperature): c = (98.6 − 32)/1.8 = 37°C. ✓

See Answers ➔

  1. Domain and range in context. Fluency

    A car park charges $4 entry plus $3 per hour (maximum 8 hours).

    • (a) Write the cost function C(h) for parking h hours.
    • (b) State the domain and range of C in context.
    • (c) Find C(5).
    • (d) How long can you park for $19?

  2. Cost, revenue and profit. Fluency

    A candle maker sells candles for $15 each. Fixed costs are $120 per batch. Variable cost is $5 per candle.

    • (a) Write functions for revenue R(n) and cost C(n).
    • (b) Write the profit function P(n).
    • (c) Find the break-even number of candles.
    • (d) State the domain of P in context (n must be a whole number ≥ 0).

  3. Function notation in context. Fluency

    The distance a car travels is d(t) = 80t km (constant 80 km/h).

    • (a) Find d(2.5) and interpret your answer.
    • (b) Find the inverse d−1(d) and interpret it in context.
    • (c) How long to travel 340 km?
    • (d) State the domain and range of d−1 in context.

  4. Composite functions in context. Fluency

    An online store converts AUD to USD using r(a) = 0.64a. Then adds 5% tax: t(u) = 1.05u.

    • (a) Find t(r(a)) — the total USD price (including tax) of an a-dollar Australian item.
    • (b) Find the total USD price of a $120 AUD item.
    • (c) Find the inverse of t(r(a)) to calculate the AUD price from a final USD price.
    • (d) If the final USD price is $67.20, what was the AUD price?

  5. Identifying model type and domain. Understanding

    A ball is dropped from 100 m. Its height h after t seconds is h(t) = 100 − 5t².

    • (a) State the model type and interpret the constants 100 and 5.
    • (b) Find the domain of h in context.
    • (c) Find the range of h.
    • (d) At what time is the ball at 55 m?

  6. Break-even with two products. Understanding

    A food truck sells burgers for $12 and chips for $5. Total fixed costs per day: $200. Variable cost: $4 per burger, $1 per chips.

    • (a) If the truck sells n burgers and 2n chips each day, write total revenue R(n) and total cost C(n).
    • (b) Find the profit function P(n) = R − C.
    • (c) Find the minimum number of burgers to break even.
    • (d) Find the profit if 40 burgers (and 80 chips) are sold.

  7. Inverse in context. Understanding

    A baker's oven preheats according to T(m) = 4m + 20 degrees Celsius, where m is minutes since turning on.

    • (a) Find T(30) and T(60).
    • (b) Find T−1(T) and interpret it in context.
    • (c) How long until the oven reaches 220°C?
    • (d) State a limitation of this model.

  8. Population with restricted domain. Understanding

    A wildlife park releases 50 animals. The population grows as P(t) = 50 × 1.12t for t years.

    • (a) Find P(0), P(5), P(10).
    • (b) The park can support at most 500 animals. After how many complete years does the population reach 500? (Use trial and error or logarithms if known.)
    • (c) What is the restricted domain once the carrying capacity is included?
    • (d) What does P(t) approach, and what does this suggest about the model long-term?

  9. Multi-step function problem. Problem Solving

    A phone plan charges: $0.30 per minute for the first 100 minutes, then $0.15 per minute after that. There is a monthly base fee of $25.

    • (a) Write a piecewise function C(m) for the monthly charge for m minutes.
    • (b) Find C(80), C(100) and C(150).
    • (c) A customer's bill is $52. How many minutes did they use?
    • (d) Sketch C(m) for 0 ≤ m ≤ 200, labelling the "kink" point.

  10. Extended: design a pricing model. Problem Solving

    A gym wants to set membership fees. Fixed costs are $8 000/month. Variable cost per member is $30/month (facilities, admin). They want to make at least $2 000 profit per month.

    • (a) Write the total cost function C(n) where n is the number of members.
    • (b) If the monthly fee is $f per member, write the profit function P(n, f) = nf − C(n).
    • (c) They currently have 200 members. What is the minimum monthly fee f to achieve $2 000 profit?
    • (d) If they charge $80/month, how many members do they need to achieve $2 000 profit?