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Mathematics and Statistics · Class 11 Commerce

Ch 2Functions — Class 11 Mathematics and Statistics, concept-first.

In an earlier chapter a relation from a set to a set was defined as any subset of the Cartesian product — that is, any chosen collection of ordered pairs with and .

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Key concepts

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Function, Domain, Codomain and Range

A function is a relation in which every element of the domain is paired with exactly one element of the codomain — both existence (no input left out) and uniqueness (no input with two images) must hold.

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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.

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From Relations to Functions — Definition, Domain, Codomain and Range

In an earlier chapter a relation from a set to a set was defined as any subset of the Cartesian product — that is, any chosen collection of ordered pairs with and .

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

Once we know a rule is a function, we classify how it matches elements of to elements of . Four standard names cover this classification.

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Even and Odd Functions

A separate and very useful way of classifying a real function looks at the symmetry of its rule — how compares with .

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Standard Functions and Their Algebraic Forms

The syllabus expects familiarity with a catalogue of standard functions, each identified by its algebraic form, its domain, its range and its graph shape.

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Exponential and Logarithmic Functions

Two further standard functions, of central importance in commercial mathematics (compound growth, present value, depreciation), are the exponential and logarithmic functions.

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Composite Functions

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

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Inverse Functions and Graphs

If is bijective (Section 2), then for every there is exactly one with . This lets us define a new function, the inverse function , by

Exercises

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  1. Q4Which of the following relations from $A=\{1,2,3\}$ to $B=\{p,q\}$ is a function? (a) $\{(1,p),(1,q),(2,p),(3,q)\}$ (b) $\{(1,p),(2,q)\}$ (c…Free
  2. Q5Find the natural domain and the range of the rational function $f(x)=\dfrac{2x}{x-4}$.Free
  3. Q6Evaluate $[3.7]$, $[-2.3]$, $[5]$ and $[-0.6]$, where $[\,\cdot\,]$ denotes the greatest integer function, and state the range of $f(x)=[x]$…Free
  4. Q7Determine whether each function is even, odd, or neither: (i) $f(x)=x^4-3x^2$ (ii) $g(x)=x^3+2x$ (iii) $h(x)=x^2+x$.Preview
  5. Q8Given $f(x)=x^2-1$ and $g(x)=x+2$, find $(f\circ g)(x)$ and $(g\circ f)(x)$, and show they are not equal.Preview
  6. Q9Using the laws of logarithms, and given $\log_{10}2=0.3010$ and $\log_{10}3=0.4771$, evaluate $\log_{10}12$ and $\log_{10}1.5$.Preview
  7. Q10Find the natural domain of $f(x)=\sqrt{x-2}+\dfrac{1}{x-5}$.Preview
  8. Q11A shopkeeper's monthly electricity bill $B$ (in rupees) is a linear function of the number of units $u$ consumed. When $200$ units are used…Preview
  9. Q12For $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^2$, which statement is correct? (a) one-one but not onto (b) onto but not one-one (c) nei…Preview
  10. Q13If $f(x)=\dfrac{x+1}{x-1}$, $x\neq 1$, show that $f$ is its own inverse, i.e. $f(f(x))=x$.Preview

More questions

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  1. Example 1Let $A=\{1,2,3,4\}$ and $B=\{1,4,9,16,25\}$. The relation $f$ from $A$ to $B$ is defined by $f(x)=x^2$. Show that $f$ is a function, and sta…Free
  2. Example 2Show that the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=3x-7$ is bijective, and hence find its inverse.Preview
  3. Example 3If $f(x)=2x+1$ and $g(x)=x^2-3$, find $(f\circ g)(x)$ and $(g\circ f)(x)$, and verify both at $x=2$.Preview