HL · Further functions 1 · Composite functions

Module 17 of 47Lesson 1 of 9 · HL · Further functions 1 · Composite functions

Composite & inverse functions, transformations, further models · Essential understanding 1

Composite functions

IB AI AHL 2.7with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: A composite function is a machine built from two machines joined end to end. The answer from the first machine is fed straight into the second. The order matters, and the second machine only works if it is given numbers it accepts.

By the end of this lesson you can:

read and write the composite function notation f(g(x)) and (f∘g)(x)
work out a composite function at a number and as a formula in x
see that f(g(x)) and g(f(x)) are usually different
solve equations such as (f∘g)(x) = k
find the domain of a composite function
work with composite functions given in a table of values

🧑🏾 Imagine this…

You buy a pair of shoes in Francistown. First the shop takes 15% off. Then it adds a delivery charge. That is two functions, one after the other: the price goes into the discount machine, and what comes out goes into the delivery machine. If the shop added the delivery charge first and took 15% off afterwards, you would pay a different amount. Same two machines, different order, different answer.

Composite function

the function you get when you put the output of g into f. It is written f(g(x)) or (f∘g)(x).

(f∘g)(x)

read “f of g of x” or “f after g”. Do g first, then f. The function written closest to x acts first.

f∘f

the function f applied twice: (f∘f)(x) = f(f(x)).

Domain of f∘g

all x that g accepts and for which g(x) is accepted by f.

★ Composite functions: the rules

(f∘g)(x) = f(g(x)): replace every x in the formula of f by the whole formula of g.
(g∘f)(x) = g(f(x)): replace every x in the formula of g by the whole formula of f.
In general f(g(x)) ≠ g(f(x)). (Sometimes they are equal, but never assume it.)
Domain of f∘g: x must be in the domain of g, and g(x) must be in the domain of f.
The range of f∘g is found by feeding the range of g into f.

⚠️ Mistakes that lose marks

Wrong order. (f∘g)(4) means g(4) first, then f of that. Many students do f first because f is written first.
f(g(x)) is not f(x) × g(x). The small circle ∘ means “after”, not “times”.
Missing brackets. If f(x) = x² and g(x) = x + 3, then f(g(x)) = (x + 3)², not x² + 3.
Forgetting the domain. √ needs a number ≥ 0 and $\dfrac{1}{}$ needs a non-zero number. Always ask: is g(x) allowed to go into f?

Prof. Kitso says

When a question gives you f(g(x)) to evaluate, work from the inside out, like peeling an onion. First the bracket nearest x, then the next one. Write the middle answer down: g(4) = … then f(…) = …. It earns method marks even if the last step goes wrong.

🧪 Try it yourselfTwo orders, two graphs

Here f(x) = ax + b and g(x) = x². The red curve is f(g(x)) = ax² + b and the blue curve is g(f(x)) = (ax + b)². Move a and b. Do the curves ever look the same? Set a = 3 and b = 2 and compare with Example 1.1 (there g(x) = x² − 1, so the curves differ a little).

Worked examples

Example 1.1Composite at a number and as a formula

f(x) = 3x + 2 and g(x) = x² − 1.
(a) Find f(g(2)) and g(f(2)). (b) Find a formula for f(g(x)) and for g(f(x)). (c) Use your formulas to check part (a).

🧠 Think it through

Inside out. For f(g(2)): do g(2) first, then put that answer into f.
For the formula, replace x in f by the whole of g(x). Use brackets.
Expand (3x + 2)² carefully: it is (3x + 2)(3x + 2).

✍️ By hand
(a) , so .
, so .
(b) .
.
(c)  ✓ and  ✓.

🔢 On your TI-Nspire CX II

On a Calculator page define both functions, then test the composites at 2.
To see the formula as a graph, open a Graphs page and type f3(x)=f1(f2(x)) with f1 = 3x+2 and f2 = x^2-1. The curve of f3 is 3x² − 1.

✅ Answer (a) f(g(2)) = 11, g(f(2)) = 63. (b) f(g(x)) = 3x² − 1; g(f(x)) = 9x² + 12x + 3.

Same two functions, two different answers: 11 and 63. This is why the order matters.

Example 1.2Discount and delivery

Kabo buys a jacket marked at P p in Francistown. The shop has two machines. The discount machine is d(p) = 0.85p (15% off). The delivery machine is c(p) = p + 35 (a P35 delivery charge).
(a) Find a formula for c(d(p)) and for d(c(p)). (b) The jacket is marked at P400. Find the two possible prices. (c) Which order is better for Kabo and why?

🧠 Think it through

c(d(p)) means discount first, then add delivery. d(c(p)) means add delivery first, then discount the total.
Substitute p = 400 into each formula.
Kabo wants the lower price.

✍️ By hand

(a)

c(d(p)) = 0.85p + 35

and

d(c(p)) = 0.85(p + 35) = 0.85p + 29.75

(b)

c(d(400)) = 0.85(400) + 35 = 375

and

d(c(400)) = 0.85(435) = 369.75

🔢 On your TI-Nspire CX II

Calculator page.

✅ Answer (a) c(d(p)) = 0.85p + 35; d(c(p)) = 0.85p + 29.75. (b) P375 or P369.75. (c) Add delivery first, then discount: P369.75.

Compare with a 15% discount followed by 14% VAT: 0.85 × 1.14 = 0.969 in either order. Two multiplications always commute, but a multiplication and an addition usually do not.

Example 1.3Solving an equation with a composite function

f(x) = 3x − 5 and g(x) = x² + 1.
(a) Find (f∘g)(x) and solve (f∘g)(x) = 19. (b) Solve (g∘f)(x) = 10.

🧠 Think it through

Build the formula first. (f∘g)(x) = f(g(x)) = 3(x² + 1) − 5.
Then it is just an equation: get x² on its own and take square roots. Remember ±.
For (g∘f)(x) put 3x − 5 inside g.

✍️ By hand

(a)

(f \circ g)(x) = 3(x^2 + 1) - 5 = 3x^2 - 2

3x^2 - 2 = 19

→

x^2 = 7

→

x = \pm\sqrt{7} = \pm 2.65

(3 s.f.).

(b)

(g \circ f)(x) = (3x - 5)^2 + 1

(3x - 5)^2 + 1 = 10

→

(3x - 5)^2 = 9

→

3x - 5 = \pm 3

x = \dfrac{8}{3}

x = \dfrac{2}{3}

🔢 On your TI-Nspire CX II

Define f and g, then solve with menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve. Give a starting guess to reach each solution.
For part (b) the same way.

✅ Answer (a) 3x² − 2; x = ±√7 = ±2.65. (b) x =

\dfrac{2}{3}

or x =

\dfrac{8}{3}

A quadratic in disguise gives two answers. If your GDC shows only one, change the starting guess.

Example 1.4The domain of a composite function

f(x) = √x (for x ≥ 0) and g(x) = x² − 9, for all real x.
(a) Find (f∘g)(x) and its largest possible domain. (b) Find (g∘f)(x) and its largest possible domain. (c) Write down the range of g∘f.

🧠 Think it through

f∘g is √(x² − 9). The square root needs x² − 9 ≥ 0.
g∘f is (√x)² − 9. But f only accepts x ≥ 0, so this restriction stays even though the formula simplifies to x − 9.
The range of g∘f: x − 9 with x ≥ 0 gives values from −9 upwards.

✍️ By hand

(a)

(f \circ g)(x) = \sqrt{x^2 - 9}

. Need

x^2 - 9 \ge 0

, so

x^2 \ge 9

: x ≤ −3 or x ≥ 3.

(b)

(g \circ f)(x) = (\sqrt{x})^2 - 9 = x - 9

, but only for x ≥ 0 (the domain of f).

🔢 On your TI-Nspire CX II

Graph f3(x) = f1(f2(x)) with f1 = sqrt(x) and f2 = x^2-9. The graph only exists for x ≤ −3 and x ≥ 3. The gap between −3 and 3 is the part of the domain that is not allowed.
📍 Where to find it on the TI-Nspire
⌨️ square root: ctrl then x² on the keypad, or type sqrt(

✅ Answer (a) √(x² − 9); x ≤ −3 or x ≥ 3. (b) x − 9; x ≥ 0. (c) (g∘f)(x) ≥ −9.

(g∘f)(x) = x − 9 looks like a straight line for all x, but the true graph is only the half-line with x ≥ 0. Always state the domain.

Example 1.5Composite functions from a table

The functions f and g both have domain {1, 2, 3, 4, 5}.

x	1	2	3	4	5
f(x)	4	1	5	2	3
g(x)	2	5	1	3	4

(a) Find (f∘g)(3) and (g∘f)(2). (b) Solve (f∘g)(x) = 5. (c) Find (g∘g)(2). (d) Find f(f(f(1))).

🧠 Think it through

No formulas here: read the table twice, inside function first.
For (b) work backwards: which input of f gives 5? Then which x makes g(x) equal to that?

✍️ By hand

(a) g(3) = 1, then f(1) = 4. So (f∘g)(3) = 4. f(2) = 1, then g(1) = 2. So (g∘f)(2) = 2.

(b) f(3) = 5, so we need g(x) = 3. From the table g(4) = 3, so x = 4. Check: g(4) = 3, f(3) = 5 ✓.

(d) f(1) = 4, f(4) = 2, f(2) = 1. So f(f(f(1))) = 1.

✅ Answer (a) 4 and 2. (b) x = 4. (c) 4. (d) 1.

In (d) the numbers 1 → 4 → 2 → 1 go round in a cycle of length 3, so applying f three times brings you home.

Challenge questions

⭐ Challenge 1.6When are the two orders equal?Paper 1 style

f(x) = 2x − 1 and g(x) = x² + 3.
(a) Find (f∘g)(x) and (g∘f)(x). (b) Solve (f∘g)(x) = (g∘f)(x), giving your answers to 3 s.f.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Both formulas first: f(g(x)) = 2(x² + 3) − 1 and g(f(x)) = (2x − 1)² + 3.
Set them equal, collect everything on one side, and use the quadratic formula or the GDC.

🔢 On your TI-Nspire CX II

Use menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve or polyRoots on 2x² − 4x − 1.

✅ Answer (a) (f∘g)(x) = 2x² + 5; (g∘f)(x) = 4x² − 4x + 4. (b) 2x² + 5 = 4x² − 4x + 4 gives 2x² − 4x − 1 = 0, so

x = 1 \pm \dfrac{\sqrt{6}}{2}

, i.e. x = 2.22 or x = −0.225.

⭐ Challenge 1.7An oil spillPaper 2 style

Oil leaks from a ship and spreads over the sea in a circle. After t minutes the radius is r(t) = 3√t metres, for t ≥ 0. The area of a circle of radius r is A(r) = πr².
(a) Find a formula for (A∘r)(t), the area after t minutes, in its simplest form. (b) Find the area after 4 minutes. (c) Find how long it takes for the area to reach 500 m². (d) Explain what you notice about (A∘r)(t) and why this happens. (e) State the domain of A∘r in this context.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Put r(t) inside A: A(r(t)) = π(3√t)².
(3√t)² = 9t. Simplify before substituting numbers.
(c) Solve 9πt = 500.
(d) Is the final formula linear or curved? Why did the square root disappear?

✍️ By hand

(a)

(A \circ r)(t) = \pi(3\sqrt{t})^2 = 9\pi t

(b)

9\pi(4) = 36\pi = 113 \text{ m}^2

(3 s.f.).

(c)

9\pi t = 500

→

t = \dfrac{500}{9\pi} = 17.7

minutes (3 s.f.).

🔢 On your TI-Nspire CX II

Calculator page.

✅ Answer (a) 9πt. (b) 113 m². (c) 17.7 minutes. (d) The area grows at a constant rate (linear in t), because the radius grows like √t and area depends on the radius squared: the square root and the square cancel. (e) t ≥ 0 (time cannot be negative; r(t) is always accepted by A).

Next: Lesson 2 · Inverse functions

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · Further functions 1 · Composite functions

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Composite functions

Worked examples

Challenge questions