Grade by Grade
D. Characteristics of Functions · Combining Functions and Modelling

Solving Problems with Composition, and Decomposing Functions

D2.6 D2.5About 75 minutes
Learning goals
  • I can build a composite function from a description and use it to answer questions.
  • I can decompose a function h into an outer function f and an inner function g so that h(x) = f(g(x)).
  • I can decide whether the order of two real-world processes changes the result.
  • I can state a sensible domain for a composite model in context.

Words to know

decompose
To write a given function h as a composition f(g(x)) of simpler functions.
composite model
A model built by feeding the output of one real-world function into another, such as area as a function of time through the radius.
HST
The Harmonized Sales Tax. In Ontario it is 13% on most purchases.
inner chunk
The part of a formula that is calculated first and then acted on, such as 3x − 1 inside (3x − 1)⁴.

Ripples on a lake

On a calm summer morning you toss a pebble into the lake at the cottage. A circular ripple spreads out. Suppose its radius grows at 30 cm each second, so r(t) = 30t. The area inside the ripple is A(r) = πr². The area depends on the radius, and the radius depends on time, so area depends on time through the radius:

A(r(t)) = π(30t)² = 900πt².

Composition is how real models are built: one quantity feeds the next. In this lesson you will build composite models from descriptions, use them to solve problems, and also go backwards: split a complicated formula into simple pieces. In Calculus and Vectors (MCV4U), splitting a function into an outer and an inner part is the first step of the chain rule.

t = 1 st = 2 st = 3 sr = 90 cmtr(t) = 30trA(r) = πr²Aseconds → centimetres → square centimetresA(r(t)) = π(30t)² = 900πt²At t = 3: A = 8100π ≈ 25 447 cm²
Time feeds the radius, and the radius feeds the area. Composing the two gives area directly as a function of time.
Worked example

Example 1: When does the ripple enclose 1 m²?

For the ripple, r(t) = 30t cm and A(r) = πr² cm². (a) Find the area after 2 s. (b) When does the area reach 1 m²?

  1. Build the composite model: A(r(t)) = π(30t)² = 900πt².
  2. (a) A(r(2)) = 900π(4) = 3600π ≈ 11 310 cm².
  3. (b) Convert units first: 1 m² = 100 cm × 100 cm = 10 000 cm².
  4. Solve 900πt² = 10 000: t² = 10 000/(900π) ≈ 3.5368, so t ≈ 1.88 s (t ≥ 0 in context).

(a) about 11 310 cm² (b) after about 1.88 s.

0.511.522.550001000015000(1.88, 10 000)A = 900πt²A (cm²)t (s)1 m² = 10 000 cm²
The composite model A = 900πt² grows like a parabola. It crosses the line A = 10 000 cm² (1 m²) at t ≈ 1.88 s.

Does the order of real processes matter?

In a store, a $10 coupon and 13% HST are two functions:

  • coupon: C(x) = x − 10
  • tax: T(x) = 1.13x

If the coupon is applied before tax, you pay T(C(x)) = 1.13(x − 10) = 1.13x − 11.30. If tax is added first and then $10 is taken off, you pay C(T(x)) = 1.13x − 10. The first is $1.30 cheaper: the coupon also saves you the tax on $10.

But a 20% discount D(x) = 0.8x and the tax commute: T(D(x)) = 1.13(0.8x) = 0.904x and D(T(x)) = 0.8(1.13x) = 0.904x. Two "multiply by a constant" functions always give the same result in either order.

Worked example

Example 2: Food left out of the fridge

The number of bacteria in a food sample is N(T) = 20T² − 80T + 500, where T is the temperature in °C, for 2 ≤ T ≤ 14. When the food is taken out of the fridge, its temperature is T(t) = 4t + 2, where t is in hours, 0 ≤ t ≤ 3. (a) Find N(T(t)). (b) When does the bacteria count reach 2000?

  1. (a) Substitute T(t) into N: N(T(t)) = 20(4t + 2)² − 80(4t + 2) + 500.
  2. Expand: 20(16t² + 16t + 4) − 320t − 160 + 500 = 320t² + 320t + 80 − 320t + 340.
  3. So N(T(t)) = 320t² + 420, for 0 ≤ t ≤ 3 (in that time T goes from 2 to 14, inside the domain of N).
  4. (b) 320t² + 420 = 2000 gives 320t² = 1580, t² = 4.9375, t ≈ 2.22 h (t ≥ 0).

N(T(t)) = 320t² + 420; the count reaches 2000 after about 2.22 hours (about 2 h 13 min).

Decomposing a function

Going backwards is just as useful. To decompose h(x), look for an inner chunk that is calculated first, and call it g(x). Whatever is done to that chunk is the outer function f.

  • h(x) = (3x − 1)⁴: inner g(x) = 3x − 1, outer f(x) = x⁴.
  • h(x) = √(x² + 9): inner g(x) = x² + 9, outer f(x) = √x.
  • h(x) = 2^(sin x): inner g(x) = sin x, outer f(x) = 2ˣ.
  • h(x) = log(5x + 2): inner g(x) = 5x + 2, outer f(x) = log x.

There is often more than one correct decomposition. For h(x) = 1/(x − 5)², both f(x) = 1/x², g(x) = x − 5 and f(x) = 1/x, g(x) = (x − 5)² work. Avoid the useless split f(x) = x, g(x) = h(x): it is correct but tells you nothing.

h(x) =(3x − 1)⁴inner: g(x) = 3x − 1outer: f(u) = u⁴ (raise the box to the 4th power)So h(x) = f(g(x)) withf(x) = x⁴ and g(x) = 3x − 1.Check: f(g(x)) = (3x − 1)⁴ ✓
Decomposing (3x − 1)⁴: the shaded box is the inner function g(x) = 3x − 1, and the dashed outer box says what is done to it, raising it to the 4th power.
Worked example

Example 3: Decompose in two ways, then check

Write h(x) = √(cos x + 2) as f(g(x)) in two different ways, and check one of them.

  1. Way 1: inner g(x) = cos x + 2, outer f(x) = √x.
  2. Way 2: inner g(x) = cos x, outer f(x) = √(x + 2).
  3. Check way 2: f(g(x)) = √(cos x + 2) ✓.
  4. Domain: cos x + 2 ≥ 1 > 0 for every x, so h is defined for all real x. Its range is √1 ≤ y ≤ √3, that is 1 ≤ y ≤ √3.

f(x) = √x with g(x) = cos x + 2, or f(x) = √(x + 2) with g(x) = cos x.

Worked example

Example 4: A Ferris wheel as a composition

A Ferris wheel at a fall fair turns once every 60 s. The angle it has turned (in radians) after t seconds is θ(t) = (π/30)t. A rider's height above the ground (m) at angle θ is h(θ) = 10 − 9 cos θ. Find h(θ(t)) and the height after 20 s.

  1. Compose: h(θ(t)) = 10 − 9 cos(πt/30).
  2. Check the period: 2π ÷ (π/30) = 60 s ✓.
  3. At t = 20: θ = 20π/30 = 2π/3, and cos(2π/3) = −1/2 (an exact value from the sibling unit Radian Measure and Trigonometric Functions).
  4. h = 10 − 9(−1/2) = 14.5 m.

h(t) = 10 − 9 cos(πt/30); after 20 s the rider is 14.5 m above the ground.

Just for fun

Gardeners decompose banana peels into compost. Mathematicians decompose (3x − 1)⁴ into x⁴ and 3x − 1. Both are useful, but only one of them makes the kitchen smell.

Hands-on activity

Coupon first or tax first?

You need: a store flyer or an online shopping page, a calculator, and paper.

  1. Choose three items with different prices (for example about $15, $60 and $200).
  2. For each item, find the total with a $10 coupon applied before 13% HST: T(C(x)) = 1.13(x − 10).
  3. Find the total if tax were added first and then $10 taken off: C(T(x)) = 1.13x − 10.
  4. Repeat with a 25% discount D(x) = 0.75x instead of the coupon, both orders.
  5. Write a one-sentence rule for when the order matters.

What to look for: with the coupon, the coupon-first total is always $1.30 less (for example $60 gives $56.50 versus $57.80), no matter the price. With the percentage discount, both orders give the same total (0.8475x; $60 gives $50.85). Rule: two "multiply by a number" steps commute; mixing "subtract a fixed amount" with "multiply" does not.

Your turn

1

A spherical balloon's radius is r(t) = 3t cm after t seconds. Its volume is V(r) = (4/3)πr³. Find V(r(t)).

2

Use the previous question. When is the volume 1000 cm³?

3

Decompose h(x) = (x² − 5)³ into f(g(x)).

4

Decompose h(x) = 10^(2x + 1).

5

Decompose h(x) = sin²x (that is, (sin x)²).

6

A $50 item: compare a $5 coupon applied before 13% HST with the coupon applied after tax.

More practice: warm-up

1

r(t) = 5t and A(r) = πr². Find A(r(1)).

2

Decompose h(x) = √(x + 7).

3

Decompose h(x) = 1/(2x − 3).

4

Decompose h(x) = log(x²).

5

Do "multiply by 1.13" and "multiply by 0.8" commute?

More practice: on your own

1

Suppose (for this question only) 1 Canadian dollar buys 0.73 US dollars and 1 US dollar buys 0.92 euros. Write U(c) and E(u), then E(U(c)), and convert $500 CAD to euros.

2

A rider's height on a Ferris wheel is h(θ) = 12 − 10 cos θ, and θ(t) = (π/45)t. Find the period of h(θ(t)) and the height at t = 45 s.

3

Two successive discounts of 20% and then 15%: is the total discount 35%? Use composition.

4

Decompose h(x) = 3 sin(2x − π) + 1 into an outer and an inner function in a way that shows the transformations.

5

For N(T(t)) = 320t² + 420 in Example 2, what is the bacteria count after 1.5 h, and why does the model stop at t = 3?

Challenge

1

Find f(x) if f(2x + 1) = 4x² + 4x + 3.

2

f(x) = x² + 1 and h(x) = 4x² − 12x + 10. Find all linear functions g with f(g(x)) = h(x).

3

An oil spill is circular. Its radius grows by r(t) = 4√t m (t in hours). Show that its area grows linearly, and find when it covers 1 hectare (10 000 m²).

Check your understanding

Quiz

1.

A ripple's radius is r(t) = 20t cm. The area is A(r) = πr². Which is A(r(t))?

2.

Which pair gives h(x) = f(g(x)) for h(x) = (x + 4)⁵?

3.

With 13% HST, T(x) = 1.13x, and a $20 coupon, C(x) = x − 20. What does T(C(x)) represent?

4.

Which pair of functions commutes (gives the same result in either order)?

5.

A Ferris wheel turns through θ(t) = (π/20)t radians in t seconds. What is the period of h(θ(t)) = 9 − 8 cos(θ(t))?

6.

Which decomposition of h(x) = 1/(x − 5)² is NOT correct?

For parents and teachers

Big idea: D2.6 asks students to solve problems involving composition, including real-world problems. This lesson also practises decomposition, which is essential for the chain rule in Calculus and Vectors (MCV4U).

How to help: when your teen meets a word problem, ask "What depends on what?" and draw the chain (time → radius → area). For decomposition, ask "What would you calculate first if you put in a number?" That part is the inner function.

Common difficulties:

  • Building the chain in the wrong order.
  • Forgetting unit conversions (1 m² = 10 000 cm², not 100 cm²).
  • Ignoring the domain in context (time cannot be negative; a model may only be valid for part of its formula's domain).
  • Thinking there is only one correct decomposition.

The exchange rates in the practice question are assumptions for the exercise, not real rates. Ontario's HST rate of 13% is real.