Functions — IB Math AA SL AA SL Study Guide

For: IB Math AA SL candidates sitting IB Math: Analysis & Approaches SL.

Covers: domain, range, composite and inverse functions, core linear/quadratic/exponential/logarithmic function properties, graph transformations, graphical equation solving, and real-world function modelling per the IB AA SL syllabus.

You should already know: IGCSE / pre-DP math.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB Math AA SL style for educational use. They are not reproductions of past IBO papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official IBO mark schemes for grading conventions.

1. What Are Functions?

A function is a rule that maps every input value from a set called the domain to exactly one output value in a set called the range, most often written as $f(x)$ where $x$ represents the input. Unlike general relations, no input can map to more than one output, a property you can verify using the vertical line test on a plotted graph: if any vertical line intersects the graph more than once, it is not a function. Functions are one of the most heavily weighted topics in IB Math AA SL, appearing across both Paper 1 (no calculator) and Paper 2 (calculator allowed), and accounting for 15-20% of total exam marks.

2. Domain, Range, Composite, Inverse

These four concepts form the foundational building blocks for all function work in the syllabus:

Domain

The domain is the set of all valid input values $x$ for which $f(x)$ is defined. Common mandatory restrictions include:

• Expressions under square roots must be non-negative
• Denominators of rational functions cannot equal zero
• Arguments of logarithmic functions must be strictly positive

Range

The range is the set of all output values $f(x)$ produced by inputs from the domain, most easily found by sketching the graph of the function or identifying bounds on the output. Worked Example 1: Find the domain and range of $f(x)=\sqrt{x+3}$.

• Domain: $x+3\ge 0⟹x\ge -3$, written as $[-3,\infty )$ in interval notation
• Range: Square roots are always non-negative, so $f(x)\ge 0$, or $[0,\infty )$

Composite Functions

A composite function applies one function to the output of another, written as $(f\circ g)(x)=f(g(x))$. Always evaluate the inner function first, and note that order matters: $f\circ g$ almost never equals $g\circ f$. Worked Example 2: If $f(x)=2x+4$ and $g(x)=x^2$, find $(f\circ g)(3)$ and $(g\circ f)(3)$.

• $(f\circ g)(3)=f(g(3))=f(9)=2(9)+4=22$
• $(g\circ f)(3)=g(f(3))=g(10)=10^2=100$

Inverse Functions

The inverse function $f^{-1}(x)$ reverses the mapping of the original function: if $f(a)=b$, then $f^{-1}(b)=a$. Inverses only exist for one-to-one functions (functions that pass the horizontal line test, with no two inputs mapping to the same output). To calculate an inverse:

1. Replace $f(x)$ with $y$
2. Swap $x$ and $y$ in the equation
3. Rearrange to solve for $y$
4. Replace $y$ with $f^{-1}(x)$ The graph of $f^{-1}(x)$ is a reflection of $f(x)$ across the line $y=x$. Worked Example 3: Find the inverse of $f(x)=5x-2$.
5. $y=5x-2$
6. $x=5y-2$
7. $5y=x+2⟹y=\frac{x+2}{5}$
8. $f^{-1}(x)=\frac{x+2}{5}$

3. Linear, Quadratic, Exponential, Log Functions

These four function types are tested in every exam sitting, either as standalone questions or embedded in calculus, statistics, or modelling problems:

Linear Functions

Form: $f(x)=mx+c$, where $m$ is the gradient (slope) and $c$ is the y-intercept. The gradient between two points $(x_1,y_1)$ and $(x_2,y_2)$ is calculated as $m=\frac{y_2-y_1}{x_2-x_1}$. The x-intercept occurs at $x=-\frac{c}{m}$ for $m\ne 0$.

Quadratic Functions

Three standard forms:

1. Standard: $f(x)=a{x}^2+bx+c$, y-intercept at $c$, axis of symmetry at $x=-\frac{b}{2a}$
2. Vertex: $f(x)=a(x-h{)}^2+k$, vertex (turning point) at $(h,k)$, opens upwards if $a>0$, downwards if $a<0$
3. Factored: $f(x)=a(x-p)(x-q)$, roots (x-intercepts) at $x=p$ and $x=q$ Worked Example: $f(x)=2(x-1{)}^2-8$ has a vertex at $(1,-8)$, opens upwards, and has roots at $x=3$ and $x=-1$.

Exponential Functions

Form: $f(x)=k{a}^x+c$, where $a>0,a\ne 1$, $k$ is the initial value when $x=0$, and there is a horizontal asymptote at $y=c$. If $a>1$ the function models growth; if $0<a<1$ it models decay.

Logarithmic Functions

The inverse of exponential functions: $y={\log }_{a}x$ is equivalent to $a^y=x$. The natural logarithm $\ln x={\log }_{e}x$, where $e\approx 2.718$ is Euler's constant. Log functions have a domain of $x>0$, a vertical asymptote at $x=0$, and pass through the point $(1,0)$. Worked Example: Solve $\ln (x)=2.5$. Rewrite in exponential form: $x=e^{2.5}\approx 12.18$.

4. Transformations of Graphs

All function transformations are applied to a parent function $f(x)$, and follow consistent rules that examiners test frequently:

1. Vertical translation: $f(x)+k$ shifts the graph up by $k$ units if $k>0$, down if $k<0$
2. Horizontal translation: $f(x-h)$ shifts the graph right by $h$ units if $h>0$, left if $h<0$ (note the opposite sign inside the bracket)
3. Vertical stretch/compression: $af(x)$ stretches the graph vertically by a factor of $a$ if $a>1$, compresses it if $0<a<1$. If $a$ is negative, reflect the graph across the x-axis first before stretching.
4. Horizontal stretch/compression: $f(bx)$ compresses the graph horizontally by a factor of $\frac{1}{b}$ if $b>1$, stretches it by $\frac{1}{b}$ if $0<b<1$. If $b$ is negative, reflect the graph across the y-axis first before stretching. Exam Tip: Always apply reflections and stretches before translations to avoid calculation errors, following the same order as BEDMAS. Worked Example: Describe the transformations mapping the parent function $f(x)=x^2$ to $g(x)=-3(x+2{)}^2+5$:
5. Reflect $f(x)$ across the x-axis
6. Stretch vertically by a factor of 3
7. Shift 2 units to the left
8. Shift 5 units upwards

5. Solving Equations Graphically

The core rule for graphical equation solving is simple: the solution to $f(x)=g(x)$ is the x-coordinate of any intersection point between the graphs of $y=f(x)$ and $y=g(x)$. If you are solving $f(x)=k$, find the intersection of $y=f(x)$ and the horizontal line $y=k$. For Paper 2, you can use your graphical display calculator (GDC) to plot both functions and use the built-in intersection tool to find solutions directly. For Paper 1, you will either be given a pre-plotted graph or be expected to sketch the function using its known properties to find intersections. Worked Example: The graphs of $y=x^2-2x-3$ and $y=2x-3$ are plotted. Find the solutions to $x^2-2x-3=2x-3$. Rearrange the equation: $x^2-4x=0⟹x(x-4)=0$, so solutions are $x=0$ and $x=4$, corresponding to the intersection points $(0,-3)$ and $(4,5)$. Exam Note: If you are asked for solutions in a specified interval, only include intersection points that fall within that interval: extra solutions outside the range will cost you marks.

6. Modelling with Functions

Functions are used to model real-world relationships across a wide range of contexts, a skill tested in 8-10 mark extended response questions on every exam:

• Linear models: Used for relationships with constant rate of change, e.g. rental costs with a fixed fee plus per-hour charge, or constant speed motion
• Quadratic models: Used for projectile motion, area optimization, or profit/loss models with a single maximum/minimum point
• Exponential models: Used for population growth, radioactive decay, compound interest, or asset depreciation
• Logarithmic models: Used for pH scales, sound intensity (decibel) measurements, or inverse exponential relationships Steps for function modelling:

1. Identify the function type from the context description
2. Use given data points to solve for unknown parameters in the function formula
3. Verify the model matches all given data points
4. Use the model to make required predictions Worked Example: A new car costs $25,000 and depreciates exponentially, with a value of $16,000 after 2 years. Find the model $V(t)=V_0{r}^t$, where $V(t)$ is the car value after $t$ years.

• Initial value $V_0=25000$, so $V(t)=25000r^t$
• Substitute $t=2,V=16000$: $16000=25000r^2⟹r^2=0.64⟹r=0.8$
• Final model: $V(t)=25000(0.8{)}^t$

7. Common Pitfalls (and how to avoid them)

1. Wrong move: Forgetting domain restrictions for log functions, e.g. writing the domain of $\ln (x-2)$ as $x\ge 2$. Why it happens: Confusing the non-negative rule for square roots with the strictly positive rule for log arguments. Correct move: Always write domains for log functions with strict inequalities, e.g. $x>2$ for this example.
2. Wrong move: Reversing the order of composite functions, e.g. calculating $f(g(x))$ as $g(f(x))$. Why it happens: Reading left to right instead of inside out. Correct move: Write brackets around the inner function to remind yourself to evaluate it first: $f(g(x))=f(\text{input }g(x))$.
3. Wrong move: Applying translations before stretches/reflections for graph transformations, leading to incorrect final equations. Why it happens: Ignoring order of operations. Correct move: Follow BEDMAS order for transformations: horizontal changes/reflections first, vertical stretches/reflections next, translations last.
4. Wrong move: Writing the inverse of $f(x)=x^2$ as $f^{-1}(x)=\sqrt{x}$ without restricting the domain. Why it happens: Forgetting that inverses only exist for one-to-one functions. Correct move: Restrict the domain of non-one-to-one functions before calculating the inverse, e.g. for $f(x)=x^2$, restrict to $x\ge 0$ so the inverse is valid.
5. Wrong move: Rounding intermediate values when solving graphical or modelling problems on Paper 2, leading to large rounding errors in the final answer. Why it happens: Trying to save time by rounding early. Correct move: Keep at least 3 significant figures for all intermediate steps, or use your GDC's stored values to avoid errors, only rounding the final answer to the required precision.

8. Practice Questions (IB Math AA SL Style)

Question 1

Let $f(x)=4-3x$ and $g(x)=\frac{2}{x-1}$, where $x\ne 1$. a) State the domain of $g(x)$ b) Calculate $(g\circ f)(2)$ c) Find the inverse function $f^{-1}(x)$

Solution

a) The denominator of $g(x)$ is zero when $x=1$, so domain is $\mathbb{R}\setminus \{1\}$ or $(-\infty ,1)\cup (1,\infty )$ b) First calculate $f(2)=4-3(2)=-2$. Then $g(-2)=\frac{2}{-2-1}=-\frac{2}{3}$ c) Swap $x$ and $y$: $x=4-3y⟹3y=4-x⟹f^{-1}(x)=\frac{4-x}{3}$

Question 2

A quadratic function has a vertex at $(2,-9)$ and passes through the point $(5,0)$. a) Write the function in vertex form $f(x)=a(x-h{)}^2+k$ b) Find all roots of $f(x)=0$ c) Describe the transformation mapping $f(x)$ to $h(x)=(x+3{)}^2+9$

Solution

a) Vertex $(2,-9)$ so $h=2,k=-9$: $f(x)=a(x-2{)}^2-9$. Substitute $(5,0)$: $0=a(3{)}^2-9⟹9a=9⟹a=1$. Final form: $f(x)=(x-2{)}^2-9$ b) Set equal to zero: $(x-2{)}^2=9⟹x-2=\pm 3⟹x=5$ and $x=-1$ c) Shift $f(x)$ 5 units left and 18 units up.

Question 3

A bacterial population grows exponentially, starting at 200 cells and reaching 1800 cells after 3 hours. a) Find the model $P(t)=P_0{a}^t$ where $t$ is time in hours b) Calculate the population after 5 hours, to the nearest whole cell c) Find the time taken for the population to exceed 10,000 cells, to 1 decimal place

Solution

a) Initial population $P_0=200$. At $t=3$: $1800=200a^3⟹a^3=9⟹a=9^{1/3}\approx 2.08$. Model: $P(t)=200(2.08{)}^t$ b) $P(5)=200(2.08{)}^5\approx 200\ast 38.29=7658$ cells c) Solve $200(2.08{)}^t>10000⟹(2.08{)}^t>50$. Take natural logs: $t>\frac{\ln 50}{\ln 2.08}\approx 5.5$ hours.

9. Quick Reference Cheatsheet

10. What's Next

Mastery of functions is foundational for almost every other topic in the IB Math AA SL syllabus. You will use function properties to solve calculus problems (differentiation and integration of all core function types), interpret statistical regression models, and solve trigonometric equations involving sine, cosine, and tangent, which are periodic functions with defined domains, ranges, and inverses. Extended response questions combining functions with calculus are some of the highest-mark questions on both Paper 1 and Paper 2, so strong foundational knowledge here will directly boost your score on later topics.

If you struggle with any of the concepts in this guide, or want to practice more exam-style questions tailored to your skill level, you can ask Ollie for personalized help, additional practice problems, or step-by-step walkthroughs of tricky transformation or modelling questions at any time on the homepage. You can also move on to our dedicated study guide for differential calculus next, where you will apply all the function rules you learned here to find gradients of curves, optimize real-world systems, and interpret rates of change.