Functions – IGCSE Additional Mathematics (0606)
1. Key Terminology (Syllabus 1.1‑1.2)
- Function – a rule that assigns to each element of a set called the domain exactly one element of another set called the range. We write the rule as \(y=f(x)\).
- Domain – the set of all permissible input values for the function.
- Range – the set of all possible output values.
- One‑to‑one (injective) function – different inputs give different outputs. Horizontal‑line test: a graph is one‑to‑one iff no horizontal line cuts it more than once.
- Many‑one function – at least two different inputs produce the same output (fails the horizontal‑line test).
- Inverse function, written \(f^{-1}(x)\) – swaps the roles of domain and range; geometrically it is the reflection of \(y=f(x)\) in the line \(y=x\).
- Composite function, written \(f\circ g\,(x)\) – apply \(g\) first, then \(f\); i.e. \(f\circ g\,(x)=f\bigl(g(x)\bigr)\).
- Product of functions, written \(fg(x)\) – simply \(f(x)\cdot g(x)\).
- Function squared, written \(f^{2}(x)\) – means \([f(x)]^{2}\); it is **not** the second iterate \(f\bigl(f(x)\bigr)\) (which would be written \(f\circ f\,(x)\)).
- Absolute‑value transformation, \(|f(x)|\) – reflects any part of the graph that lies below the \(x\)-axis above the axis.
2. Determining Domain and Range
Use the following checklist (Cambridge “Domain‑range” requirement):
| Type of function | Domain condition | Range comment |
|---|
| Rational \(f(x)=\dfrac{P(x)}{Q(x)}\) |
Exclude values that make \(Q(x)=0\). |
Often all real numbers except those excluded. |
| Even‑root radical \(f(x)=\sqrt[n]{R(x)}\) with \(n\) even |
Require \(R(x)\ge0\). |
Output is ≥ 0 (or ≤ 0 for even‑root of a negative‑leading expression). |
| Logarithmic \(f(x)=\log_a\!R(x)\) ( \(a>0,\ aeq1\) ) |
Require \(R(x)>0\). |
Range is all real numbers \(\mathbb{R}\). |
| Trigonometric – e.g. \(\tan x\) |
Exclude angles where the function is undefined (e.g. \(\frac{\pi}{2}+k\pi\)). |
Range depends on the function (e.g. \(\tan x\) has range \(\mathbb{R}\)). |
Examples
- \(f(x)=\dfrac{1}{x-2}\) → Domain \(xeq2\); Range \(yeq0\).
- \(g(x)=\sqrt{5-x}\) → Domain \(x\le5\); Range \(g(x)\ge0\).
- \(h(x)=\ln(x+3)\) → Domain \(x>-3\); Range \(\mathbb{R}\).
- \(k(x)=\tan x\) (principal period) → Domain \(xeq\frac{\pi}{2}+n\pi\); Range \(\mathbb{R}\).
3. Function Notation at a Glance
| Notation | Meaning | Typical example |
|---|
| \(f(x)\) |
Value of \(f\) at the argument \(x\) |
If \(f(x)=2x+3\), then \(f(4)=11\). |
| \(f^{-1}(x)\) |
Inverse function – swaps domain and range |
If \(f(x)=2x+3\), then \(f^{-1}(x)=\dfrac{x-3}{2}\). |
| \(fg(x)\) |
Product of two functions: \(f(x)\cdot g(x)\) |
\(f(x)=x+1,\;g(x)=x-2\;\Rightarrow\;fg(x)=(x+1)(x-2).\) |
| \(f\circ g\,(x)\) |
Composite function: \(f\bigl(g(x)\bigr)\) |
\(f(x)=x^{2},\;g(x)=2x+1\;\Rightarrow\;f\circ g\,(x)=(2x+1)^{2}.\) |
| \(f^{2}(x)\) |
Square of the function value: \([f(x)]^{2}\) |
\(f(x)=x+3\;\Rightarrow\;f^{2}(x)=(x+3)^{2}.\) |
| \(|f(x)|\) |
Absolute‑value transformation |
\(f(x)=x-2\;\Rightarrow\;|f(x)|=|x-2|.\) |
4. Using Function Notation (Typical Exam Tasks)
- Evaluate \(f(x)\) for a given \(x\).
- Find the domain and range of a given function.
- State whether a function is one‑to‑one; if not, explain why it has no inverse.
- Find the inverse function \(f^{-1}(x)\) (when it exists).
- Calculate the product \(fg(x)\) or the composite \(f\circ g\,(x)\). Emphasise that order matters for composition.
- Interpret \(f^{2}(x)\) correctly as \([f(x)]^{2}\).
- Sketch a function together with its inverse to illustrate the reflection about \(y=x\).
5. Finding the Inverse Function
- Write the relation as \(y=f(x)\).
- Interchange \(x\) and \(y\) (reflect across the line \(y=x\)).
- Solve the new equation for \(y\); the resulting expression is \(f^{-1}(x)\).
Example 1 – Linear
\[
\begin{aligned}
y &= 3x-5 \\
\text{Swap }x\text{ and }y &: \; x = 3y-5 \\
3y &= x+5 \\
y &= \frac{x+5}{3}
\end{aligned}
\qquad\Longrightarrow\qquad f^{-1}(x)=\frac{x+5}{3}
\]
Example 2 – Trigonometric (restricted domain)
\[
\begin{aligned}
y &= \sin x ,\qquad -\frac{\pi}{2}\le x\le\frac{\pi}{2}\\
\text{Swap }x\text{ and }y &: \; x = \sin y \\
y &= \arcsin x \quad\text{(principal value)}\\
\end{aligned}
\qquad\Longrightarrow\qquad f^{-1}(x)=\arcsin x
\]
6. Why Some Functions Have No Inverse
A function fails to have an inverse when it is not one‑to‑one. The horizontal‑line test provides a quick visual check: if any horizontal line meets the graph more than once, the function is not invertible on that whole domain.
Counter‑example: \(f(x)=x^{2}\) for all real \(x\).
Horizontal line \(y=4\) cuts the graph at \(x=-2\) and \(x=2\); therefore \(f\) is many‑one and has no inverse on \(\mathbb{R}\). An inverse exists only after restricting the domain (e.g. \(x\ge0\)).
7. Composite Functions
- \(f\circ g\,(x)=f\bigl(g(x)\bigr)\) – apply \(g\) first.
- \(g\circ f\,(x)=g\bigl(f(x)\bigr)\) – generally different from \(f\circ g\).
Example – Order matters
\[
\begin{aligned}
f(x)&=2x+3,\qquad g(x)=x^{2}\\[2mm]
f\circ g\,(x)&=2x^{2}+3,\\
g\circ f\,(x)&=(2x+3)^{2}=4x^{2}+12x+9.
\end{aligned}
\]
8. Product of Functions
Simply multiply the two expressions:
\[
(fg)(x)=f(x)\,g(x).
\]
Example
\[
f(x)=x+2,\;g(x)=x-3\;\Longrightarrow\;fg(x)=(x+2)(x-3)=x^{2}-x-6.
\]
9. Function Squared – \(f^{2}(x)\)
\(f^{2}(x)\) means \([f(x)]^{2}\). It is **not** the same as the second iterate \(f\bigl(f(x)\bigr)\), which would be written \(f\circ f\,(x)\).
Example
\[
f(x)=\sqrt{x}\;\Longrightarrow\;f^{2}(x)=\bigl(\sqrt{x}\bigr)^{2}=x.
\]
10. Absolute‑Value Transformation \(|f(x)|\)
Replacing \(f(x)\) by \(|f(x)|\) reflects any portion of the graph that lies below the \(x\)-axis to an equal position above the axis.
Example
\[
f(x)=x-2\quad\Rightarrow\quad |f(x)|=|x-2|.
\]
The graph of \(|x-2|\) is a V‑shaped curve obtained by reflecting the part of \(y=x-2\) that is negative (i.e. for \(x<2\)) upward.
11. Sketching a Function and Its Inverse
- Draw the graph of \(y=f(x)\) on a set of axes.
- Draw the line \(y=x\) (a 45° diagonal).
- Reflect each point of the original graph across the line \(y=x\); the reflected curve is the graph of \(y=f^{-1}(x)\).
Worked example – linear
- Function: \(f(x)=2x+1\). Plot the line.
- Reflect in \(y=x\). The reflected line has equation \(y=\frac12x-\frac12\), which is exactly \(f^{-1}(x)\).
12. Worked Examples (All Notations)
- Evaluation & inverse
\(f(x)=4x-7\)
• \(f(3)=4(3)-7=5\).
• Solve \(y=4x-7\) for \(x\): \(x=\dfrac{y+7}{4}\) → \(f^{-1}(x)=\dfrac{x+7}{4}\).
- Product of functions
\(g(x)=x^{2}+1,\;f(x)=2x-5\)
\(fg(x)=(2x-5)(x^{2}+1)=2x^{3}+2x-5x^{2}-5.\)
- Composite function
\(h(x)=3x+2\)
\(h\circ h\,(x)=h\bigl(3x+2\bigr)=3(3x+2)+2=9x+8.\)
- Function squared & self‑inverse
\(p(x)=\dfrac{1}{x}\)
\(p^{2}(x)=\left(\dfrac{1}{x}\right)^{2}=\dfrac{1}{x^{2}}\).
To find the inverse, swap \(x\) and \(y\): \(x=\dfrac{1}{y}\Rightarrow y=\dfrac{1}{x}\); thus \(p^{-1}(x)=\dfrac{1}{x}=p(x)\) (self‑inverse).
- Domain & range practice
\(q(x)=\dfrac{2}{x+1}\) → Domain \(xeq-1\); Range \(yeq0\).
\(r(x)=\sqrt{3-x}\) → Domain \(x\le3\); Range \(r(x)\ge0\).
- One‑to‑one test
\(s(x)=x^{2}\) (domain \(\mathbb{R}\)) – not one‑to‑one (fails horizontal‑line test).
\(t(x)=\ln x\) (domain \(x>0\)) – one‑to‑one (passes the test).
13. Practice Questions
- Given \(f(x)=4x-7\):
- Find \(f(3)\).
- Find the inverse function \(f^{-1}(x)\).
- If \(g(x)=x^{2}+1\) and \(f(x)=2x-5\), compute the product \(fg(x)\).
- Let \(h(x)=3x+2\). Write and simplify the composite \(h\circ h\,(x)\).
- For \(p(x)=\dfrac{1}{x}\):
- Find \(p^{2}(x)\).
- Find \(p^{-1}(x)\) and comment on its relationship to \(p(x)\).
- Determine the domain and range of each function:
- \(q(x)=\dfrac{2}{x+1}\)
- \(r(x)=\sqrt{3-x}\)
- State whether each function is one‑to‑one; justify using the horizontal‑line test:
- \(s(x)=x^{2}\) (domain \(\mathbb{R}\)).
- \(t(x)=\ln x\) (domain \(x>0\)).
- Sketch \(y=f(x)=2x+1\) together with its inverse on the same axes. Label the line \(y=x\).
14. Answers to Practice Questions
| Q. | Answer |
|---|
| 1 |
\(f(3)=5\); \(f^{-1}(x)=\dfrac{x+7}{4}\) |
| 2 |
\(fg(x)= (2x-5)(x^{2}+1)=2x^{3}+2x-5x^{2}-5\) |
| 3 |
\(h\circ h\,(x)=9x+8\) |
| 4 |
\(p^{2}(x)=\dfrac{1}{x^{2}}\); \(p^{-1}(x)=\dfrac{1}{x}=p(x)\) (self‑inverse) |
| 5 |
\(q(x)=\dfrac{2}{x+1}\): Domain \(xeq-1\); Range \(yeq0\).
\(r(x)=\sqrt{3-x}\): Domain \(x\le3\); Range \(r(x)\ge0\). |
| 6 |
\(s(x)=x^{2}\) – not one‑to‑one on \(\mathbb{R}\) (fails horizontal‑line test).
\(t(x)=\ln x\) – one‑to‑one on its domain \(x>0\) (passes the test). |
| 7 |
Graph of \(y=2x+1\) (straight line). Its inverse is the reflection across \(y=x\): \(y=\frac12x-\frac12\). Both lines intersect at \((-1,-1)\) and are symmetric about \(y=x\). |