IB Mathematics AI SL Concept of a function, domain, range and graph MAI Study Notes - New Syllabus

For a function \( f: X \rightarrow Y \),

The set of all \( x \)’s involved is called DOMAIN
The set of all \( y \)’s involved (only the images) is called RANGE

Consider again the function \( f: X \rightarrow Y \) given by

Then

DOMAIN: \( x \in X = \{1, 2, 3\} \)
RANGE: \( y \in \{a, b, d\} \)

We usually denote the domain by \( D_f \) and the range by \( R_f \).

GRAPH

DOMAIN:

Projection on the \( x \)-axis, i.e. \( D_f: x \in [1, 8] \)

RANGE:

Projection on the \( y \)-axis, i.e. \( R_f: y \in [2, 6] \)

NOTE:

The graph also shows if we have a function or not

This is not a function, since \( f(3) \) for example is not unique!

Vertical line test:

Any vertical line intersects the graph at most once.

AN “AGREEMENT” FOR THE DOMAIN

Usually, a function is simply given as a formula of the form \( y = f(x) \), where \( x \) and \( y \) are real variables.

If the domain of the function is not given, we agree that

\( D_f \text{ is } \mathbb{R} \)

or \( D_f \) is the largest possible subset of \( \mathbb{R} \)

For example,

 if \( f \) is given by \( f(x) = 2x \), we assume that \( x \in \mathbb{R} \)
 if \( f \) is given by \( f(x) = \frac{2}{x} \), we assume that \( x \in \mathbb{R} – \{0\} = \mathbb{R} \)
(we may also write \( D_f: x \neq 0 \))

1. \( f(x) \) is a function with no restrictions on \( x \),

for example a polynomial [say \( f(x) = 2x^3 + 3x^2 + 1 \)], then

\( D_f = \mathbb{R} \)

2. \( f(x) = \frac{A}{B} \), then \( B \) cannot be 0, thus

\( D_f = \mathbb{R} – \{\text{roots of the equation } B = 0\} \)

3. \( f(x) = \sqrt{A} \), then \( A \geq 0 \).

\( D_f = \text{the solution set of the inequality } A \geq 0 \)

4. \( f(x) = \log A \) or \( f(x) = \ln A \), then \( A > 0 \).¹

\( D_f = \text{the solution set of the inequality } A > 0 \)