Functions and Transformations

What Is a Function?

A function $f$ is a rule that assigns to each input value (from the domain) exactly one output value (in the range). We write $f: x \mapsto f(x)$ or simply $y = f(x)$.

The vertical line test is a quick visual check: if any vertical line intersects a graph more than once, the graph does not represent a function.

Domain and Range

The domain of a function is the set of all permitted input values ($x$-values). The range is the set of all possible output values ($y$-values).

Common domain restrictions include:

• Division by zero: for $f(x) = \frac{1}{x-2}$, we need $x \neq 2$
• Square roots of negatives: for $f(x) = \sqrt{x+3}$, we need $x \geq -3$
• Logarithms of non-positive numbers: for $f(x) = \ln(x)$, we need $x > 0$

To find the range, consider the behaviour of the function — its maximum and minimum values, asymptotes, and end behaviour. Sketching the graph (by hand or on your GDC) is often the most efficient approach.

Composite Functions

The composite function $(f \circ g)(x) = f(g(x))$ means "apply $g$ first, then $f$." The order matters: $f(g(x)) \neq g(f(x))$ in general.

To evaluate $f(g(x))$, replace every $x$ in the formula for $f$ with the expression $g(x)$.

Example: If $f(x) = 2x + 1$ and $g(x) = x^2$, then:

• $f(g(x)) = f(x^2) = 2x^2 + 1$
• $g(f(x)) = g(2x+1) = (2x+1)^2 = 4x^2 + 4x + 1$

The domain of $f \circ g$ is the set of $x$-values in the domain of $g$ for which $g(x)$ is in the domain of $f$.

Inverse Functions

The inverse function $f^{-1}$ reverses the effect of $f$. If $f(a) = b$, then $f^{-1}(b) = a$. Formally, $f(f^{-1}(x)) = f^{-1}(f(x)) = x$.

A function has an inverse if and only if it is one-to-one (injective) — each output comes from exactly one input. Graphically, this means it passes the horizontal line test.

To find the inverse algebraically:

1. Write $y = f(x)$
2. Swap $x$ and $y$
3. Solve for $y$
4. Write $f^{-1}(x) = y$

Graphical property: The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$.

Key relationship: The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

Self-Inverse Functions

A function is self-inverse if $f^{-1}(x) = f(x)$, which means $f(f(x)) = x$. Examples include $f(x) = \frac{1}{x}$ and $f(x) = -x$. The graph of a self-inverse function is symmetric about the line $y = x$.

Graph Transformations

Transformations modify the graph of a function. The IB formula booklet includes these, but understanding the logic is key.

Translations (Shifts)

Transformation	Effect on graph	Example
$y = f(x) + a$	Vertical shift up by $a$ units	$y = x^2 + 3$ shifts parabola up 3
$y = f(x) - a$	Vertical shift down by $a$ units	$y = x^2 - 2$ shifts parabola down 2
$y = f(x - a)$	Horizontal shift right by $a$ units	$y = (x-1)^2$ shifts right 1
$y = f(x + a)$	Horizontal shift left by $a$ units	$y = (x+4)^2$ shifts left 4

The crucial insight: horizontal transformations are "opposite to what you might expect." The expression $f(x - 3)$ shifts the graph right, not left. Think of it this way: to get the same $y$-value, $x$ must be 3 units larger.

Using vector notation, $y = f(x-a) + b$ is a translation by the vector $\begin{pmatrix} a \\ b \end{pmatrix}$.

Reflections

Transformation	Effect on graph
$y = -f(x)$	Reflection in the $x$-axis (multiply all $y$-values by $-1$)
$y = f(-x)$	Reflection in the $y$-axis (replace $x$ with $-x$)

Stretches (Dilations)

Transformation	Effect on graph
$y = af(x)$	Vertical stretch by factor $a$ (multiply all $y$-values by $a$)
$y = f(bx)$	Horizontal stretch by factor $\frac{1}{b}$ (compress by factor $b$)

Again, horizontal stretches are "opposite": $y = f(2x)$ compresses the graph horizontally by a factor of 2 (everything happens twice as fast).

Order of Transformations

When multiple transformations are combined, the order matters. A useful approach is to think of the transformation as acting on the $x$ or $y$ variable:

1. Inside the function (affecting $x$): Apply in reverse order — stretches first, then translations.
2. Outside the function (affecting $y$): Apply in the written order — stretches first, then translations.

For example, $y = 2f(x - 3) + 1$:

• Horizontal shift right 3 (inside)
• Vertical stretch by factor 2 (outside)
• Vertical shift up 1 (outside)

Tip 1: Sketch Graphs on Your GDC

When unsure about a transformation, graph both the original and transformed function on your GDC to visually confirm the effect. This builds intuition and catches algebraic errors.

Tip 2: Track Key Points Through Transformations

Choose 3-4 key points on the original graph (intercepts, vertex, endpoints). Apply the transformation to each point. Plot the new points and connect them. This method is reliable and scores well in exams.

Tip 3: Remember the "Opposite" Rule for Horizontal Changes

Anything that happens inside the function (affecting $x$) works in the opposite direction to what you might expect. $f(x-3)$ shifts right, $f(2x)$ compresses. Train yourself to pause and think about horizontal transformations.

Tip 4: Use the Swap Method for Inverses

The "swap $x$ and $y$, then solve" method is the most reliable way to find inverse functions. Always verify by checking that $f(f^{-1}(x)) = x$.

Tip 5: State Domain Restrictions Explicitly

In IB exams, marks are often awarded for stating the domain and range. Do not skip this step, especially for inverse functions where the domain of $f^{-1}$ equals the range of $f$.