Relations and Functions Class 11: notes and important questions

Relations and Functions: revision notes

1. Ordered pairs and Cartesian products

\((a, b) = (c, d)\) exactly when \(a = c\) and \(b = d\). The Cartesian product \(A \times B = \{(a, b) : a \in A, b \in B\}\); \(n(A \times B) = n(A) \times n(B)\). In general \(A \times B \neq B \times A\). \(\mathbb R \times \mathbb R\) is the plane.

2. Relations

A relation \(R\) from \(A\) to \(B\) is any subset of \(A \times B\). Its domain is the set of first elements, its range the set of second elements, and \(B\) is the codomain. If \(n(A) = p\) and \(n(B) = q\), there are \(2^{pq}\) relations from \(A\) to \(B\).

3. Functions

A function \(f : A \to B\) is a relation in which every element of \(A\) has exactly one image in \(B\). Domain \(A\); range \(= \{f(x) : x \in A\} \subset B\).

4. Real functions: domain and range

• Domain: all real \(x\) for which the formula makes sense: denominators \(\neq 0\), expressions under \(\sqrt{\ }\) \(\ge 0\).
• Identity \(f(x) = x\); constant \(f(x) = c\); polynomial; rational \(\dfrac{p(x)}{q(x)}\), \(q(x) \neq 0\).
• Modulus \(|x|\): \(x\) if \(x \ge 0\), \(-x\) if \(x \lt 0\); range \([0, \infty)\).
• Signum: \(1\) for \(x \gt 0\), \(0\) at \(0\), \(-1\) for \(x \lt 0\); range \(\{-1, 0, 1\}\).
• Greatest integer \([x]\): the greatest integer \(\le x\); \([3.7] = 3\), \([-2.5] = -3\); range \(\mathbb Z\).

5. Algebra of real functions

On the common domain: \((f \pm g)(x) = f(x) \pm g(x)\), \((fg)(x) = f(x)g(x)\), \(\left(\dfrac fg\right)(x) = \dfrac{f(x)}{g(x)}\) where \(g(x) \neq 0\); \((\alpha f)(x) = \alpha f(x)\).

Worked example 1

Find the domain of \(f(x) = \dfrac{x}{x^2 - 9}\).

\(x^2 - 9 \neq 0\): domain \(\mathbb R - \{-3, 3\}\).

Worked example 2

Find the range of \(f(x) = 3 - x^2\).

\(x^2 \ge 0\), so \(f(x) \le 3\): range \((-\infty, 3]\).

Worked example 3

\(A = \{1, 2\}\), \(B = \{x, y, z\}\). How many relations are there from \(A\) to \(B\)?

\(n(A \times B) = 6\), so \(2^6 = 64\) relations.

Common errors

• Calling a relation a function when some element of the domain has two images, or none.
• \([-2.5] = -2\): the greatest integer function goes down, to \(-3\).
• Cancelling \(x - 2\) in \(\dfrac{x^2 - 4}{x - 2}\) and forgetting that \(2\) is still not in the domain.
• Mixing up codomain and range.

Exam tips

• For a range, write \(y = f(x)\), solve for \(x\) and find the \(y\) for which a valid \(x\) exists.
• State the domain with every quotient \(\dfrac fg\).