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Business Mathematics and Statistics · Class 12 Commerce

Ch 4Functions — Class 12 Business Mathematics and Statistics, concept-first.

Before a relation or a function can be defined precisely, we need a way of pairing up elements from two sets so that the order in which they are written matters. An ordered pair consists of a first element and a second element , written in that fixed order; two ordered pairs and are equal only when and .

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Key concepts

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Relations and Functions

A relation from set to set is any subset of the Cartesian product — a chosen collection of ordered pairs connecting some elements of to some elements of .

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Ordered Pairs and the Cartesian Product

Before a relation or a function can be defined precisely, we need a way of pairing up elements from two sets so that the order in which they are written matters.

2

Relations — Domain and Range

A relation from a set to a set is simply any subset of the Cartesian product , i.e. . In plain terms, a relation is a chosen collection of ordered pairs that connects some elements of to some elements…

3

Functions — Definition, Domain, Codomain and Range

A function is a special kind of relation, obeying one extra rule that makes it far more useful for describing dependable, predictable relationships — such as how a firm's total cost depends on the qua…

4

Types of Functions — One-One, Onto, Into and Bijective

Once we know a rule is a function, we can further classify how it matches elements of to elements of .

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Kinds of Functions by Algebraic Form

Beyond classifying a function by how it matches elements (Section 4), the Business Mathematics Statistics syllabus also expects familiarity with several standard families of functions, identified by…

6

Composite Functions and Inverse Functions

Composite function. Given two functions and , the composite function (read 'g composed with f', or 'g of f') is defined by first applying and then applying to the result:

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Business Functions — Cost, Revenue and Profit

The whole point of studying functions in a Business Mathematics Statistics syllabus is to use them to model real economic quantities.

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Business Functions — Demand and Supply

Two further functions describe how the quantity of a good bought or sold reacts to its price, and together they let us calculate the market equilibrium.

Exercises

+Show 7 questions7 questions
  1. Q7Which of the following relations from $A=\{1,2,3\}$ to $B=\{a,b\}$ represents a function? (a) $\{(1,a),(1,b),(2,a),(3,b)\}$ (b) $\{(1,a),(2,…Free
  2. Q8Consider $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$. Which of the following correctly describes $f$? (a) One-one but not onto (b…Free
  3. Q9Find the domain and range of the rational function $f(x) = \dfrac{1}{x-2}$.Free
  4. Q10Find the domain and range of the modulus function $f(x) = |x|$, $x \in \mathbb{R}$, and sketch its graph.Preview
  5. Q11If $f(x) = x^2 - 4$ and $g(x) = x+3$, find $(f \circ g)(x)$ and $(g \circ f)(x)$, and show that $(f \circ g)(x) \neq (g \circ f)(x)$.Preview
  6. Q12A firm's profit function is $P(x) = -2x^2 + 120x - 800$, where $x$ is the number of units sold. Find the number of units that maximises prof…Preview
  7. Q13A manufacturer's total cost is a linear function of output. When output is 10 units, total cost is ₹800; when output is 20 units, total cost…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 8 questions8 questions
  1. Q1(k) If $f(x) = 3x + 7$, then $f(2)$ is equal to (a) 12 (b) 13 (c) 14 (d) 15Preview
  2. Q2(g) Show that $f(x) = 7x^6 + 3x^4 - 2x^2 + 4$ is an even function.Preview
  3. Q3Fill in the blanks : If $f(-x) = -f(x)$, then $f(x)$ is a ______ function.Preview
  4. Q4Rectify the underlined portions of the following sentences : $y = f(x) = 10$ is a _linear_ function.Preview
  5. Q5What is an inverse function ?Preview
  6. Q6What do you mean by domain of a function ?Preview
  7. Q7From the following the one which is an odd function, is : (a) $2x^3 - 3$ (b) $3x^4 - 3x^2 + 1$ (c) $7x^2 - 11$ (d) $2x^3 + 3x^4 + x^2 + 4$Preview
  8. Q8What type of function, $\{(1, 1), (2, 2), (3, 3)\}$ is ?Preview

More questions

+Show 6 questions6 questions
  1. Example 1Given $A=\{1,2,3\}$ and $B=\{4,5,6\}$, and the relation $R=\{(1,4),(2,5),(3,6)\}$ from $A$ to $B$, show that $R$ is a function and state its…Free
  2. Example 2Show that $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x+3$ is a bijective function.Free
  3. Example 3If $f(x) = x+2$ and $g(x) = 3x-1$, find $(f \circ g)(x)$ and $(g \circ f)(x)$, and verify your results at $x=2$.Preview
  4. Example 4Find the inverse of the function $f(x) = 3x - 5$, and verify that $f(f^{-1}(x)) = x$.Preview
  5. Example 5A firm's cost function is $C(x) = 50x + 2000$ and its revenue function is $R(x) = 80x$, where $x$ is the number of units produced and sold.…Preview
  6. Example 6The demand function for a product is $Q_d = 100 - 2P$ and the supply function is $Q_s = -20 + 3P$, where $P$ is the price per unit. Find the…Preview