D. Characteristics of Functions · Combining Functions and Modelling

Combining Functions and Modelling: Unit Review

D2.1 D2.2 D2.3 D2.4 D2.5 D2.6 D2.7 D2.8 D3.1 D3.2 D3.3About 40 minutes

Learning goals

I can find and describe sums, differences, products, quotients and compositions of functions.
I can use inverses and transformations as compositions.
I can compare function families, solve equations without algebra, and build models of real situations.

Words to know

composition
f(g(x)): apply g first, then f.

domain of a combined function
Where every part is defined (and, for f ∘ g, where g(x) is in the domain of f).

bisection
Halving an interval with a sign change, again and again, to close in on a solution.

Let's review!

In this unit you learned to treat functions like numbers: you added, subtracted, multiplied and divided them and found the key features of the results; you tested combined functions for symmetry and increasing behaviour; and you met real combinations such as a damped spring, a phone keypad tone and profit. Then you composed functions with tables, graphs and algebra, found domains and ranges of compositions, decomposed functions, saw that a function composed with its inverse gives x, and recognized every transformation as a composition with a linear function. Finally, you compared all the function families of the course, solved equations and inequalities that algebra cannot reach, and built models of real situations.

This is the last unit of Advanced Functions (MHF4U). If you are going on to Calculus and Vectors (MCV4U), composition and decomposition are the heart of the chain rule, and the function families are the functions you will differentiate. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do two or three sections at a time. Keep a calculator in radian mode for any question with trigonometric functions. For every combined or composite function, find the domain first. Sketch a quick graph whenever you can: it catches most errors with signs, asymptotes and the number of solutions. For each question you miss, write down why (wrong order of composition? lost a restriction? wrong family?) and go back to that lesson. Finish with the 10-question quiz.

How the unit fits together. Lesson numbers are in brackets: combining functions (green), composition (blue), and comparing, solving and modelling (yellow).

Just for fun

You can now model a swing slowly stopping, decode a phone keypad tone, decide whether to use your coupon before tax, and solve cos x = x with one finger on a calculator. Not bad for a unit that started with adding two graphs.

Sums and Differences of Functions

f(x) = x² − 4 and g(x) = 2x + 1. Find (f − g)(x) and its zeros.

State the domain of h(x) = √(x + 3) + log(5 − x).

The graphs of f and g meet at x = −2 and x = 6. Where is f − g equal to 0?

Products of Functions

f(x) = x − 1 and g(x) = (x + 2)². Find the zeros of fg and say where the graph only touches the axis.

Between which lines does y = 2x sin x lie?

Find the domain and range of y = √x · √(6 − x).

Quotients of Functions

Find the domain, vertical asymptote and hole of y = (x² − 1)/(x² − 4x + 3).

What is the range of y = 2/(x² + 4)?

Why does (sin x)/x have a hole, not an asymptote, at x = 0?

Properties of Combined Functions: Even, Odd, Increasing, Decreasing

Is y = x² sin x even, odd or neither?

Is y = 3ˣ + 3⁻ˣ even, odd or neither?

Show that 2ˣ + x³ = 20 has exactly one solution.

Real-World Combinations: Damped Motion, Tones and Profit

For d(t) = 15(0.85)ᵗ cos(πt), find when the amplitude falls below 5 cm.

Which two tones make up the sound of key 3 on a phone keypad?

R(x) = 20x − 0.1x² and C(x) = 600 + 4x. Find the maximum profit.

Composition of Functions: Tables, Machines and Graphs

g(3) = 7 and f(7) = −2. Find (f ∘ g)(3).

x: 0, 1, 2 with f: 2, 0, 1 and g: 1, 2, 0. Find f(g(0)) and g(f(0)).

A(r) is the area of a pizza of radius r (cm), and r(d) = d/2 for diameter d. What does A(r(30)) mean?

Composition Algebraically: Equations, Domain and Range

f(x) = 2x − 3 and g(x) = x². Find f(g(x)) and g(f(x)).

f(x) = √(x − 2) and g(x) = x + 5. State the domain of f(g(x)).

f(x) = 1/x and g(x) = x − 4. Find g(f(x)) and its range.

What is the range of y = 3^(cos x)?

Solving Problems with Composition, and Decomposing Functions

Decompose h(x) = √(4x − 1) and k(x) = sin(x²).

A balloon's radius is r(t) = 2t + 1 cm. Write V(r(t)) using V(r) = (4/3)πr³ and find V at t = 1.

A $40 item: $5 coupon before 13% tax, or after?

Composing a Function with Its Inverse

Show that f(x) = (x + 4)/2 and g(x) = 2x − 4 are inverses.

Simplify 10^(log 13) and log(10^(−2.5)).

Evaluate sin⁻¹(sin(5π/6)).

Transformations as Composition with Linear Functions

Write y = sin(4x − π) as f(A(x + B)) and give the phase shift.

Under f(g(x)) with g(x) = 3(x + 2), where does the point (6, 1) go?

Describe g(f(x)) for f(x) = log x and g(x) = −2(x − 1).

Comparing the Function Families

x = 0, 1, 2, 3 and y = 7, 14, 28, 56. Which family?

Which families have vertical asymptotes?

Order from slowest to fastest growth: x³, log x, 1.1ˣ.

When Algebra Is Not Enough: Solving Graphically and Numerically

h(1) = −0.4 and h(2) = 1.1 for a continuous h. After one bisection step h(1.5) = 0.3. Which interval now holds the zero?

Solve 2ˣ ≤ x + 3 using the solutions x ≈ −2.86 and x ≈ 2.44 of 2ˣ = x + 3.

How many solutions does cos x = x/3 have? (Hint: |cos x| ≤ 1.)

Building and Using Function Models

A drink cools from 80°C in a 20°C room; after 10 min it is 50°C. Assuming T = 20 + 60bᵗ, find b.

A water depth varies from 1.5 m to 4.5 m with period 12 h, deepest at t = 3. Write a model.

Name one limitation of an exponential model for the number of users of a new app.

Check your understanding

Quiz

What is the domain of (f + g)(x) if f(x) = √x and g(x) = 1/(x − 2)?

Which function has a hole at x = 3?

f is odd and g is odd. What is f · g?

Which model describes a spring whose bounces shrink by 20% every second?

f(x) = x² + 1 and g(x) = x − 3. What is f(g(x))?

What is the domain of f(g(x)) for f(x) = log x and g(x) = x − 7?

Which pair of functions are inverses?

What is the horizontal translation in y = cos(2x + π/3)?

A table with equally spaced x-values has y-values 3, 6, 12, 24, 48. Which family fits?

10.

Which equation must be solved graphically or numerically?

For parents and teachers

This review covers the whole unit. Use mistakes to decide what to revisit:

Domain errors with sums, products or quotients → Sums and Differences of Functions and Quotients of Functions.
Sign or zero errors in products → Products of Functions.
Even/odd or increasing mistakes → Properties of Combined Functions.
Trouble setting up real combinations (damped motion, profit) → Real-World Combinations.
Composing in the wrong order → Composition of Functions: Tables, Machines and Graphs.
Algebra, domain or range of f(g(x)) → Composition Algebraically.
Word problems with composition, or decomposing → Solving Problems with Composition.
Inverse and 10^(log x) restrictions → Composing a Function with Its Inverse.
Phase shifts and mapping rules → Transformations as Composition with Linear Functions.
Choosing the wrong family → Comparing the Function Families.
Missing solutions or wrong accuracy → When Algebra Is Not Enough.
Models without units, context or limits → Building and Using Function Models.

A good final check: ask your teen to explain, in their own words, the difference between f(x)·g(x) and f(g(x)), with an example of each.