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CBSE · Class 11 · Mathematics

Functions (Ch 2: 2.4)

4 concepts · 32 practice questions with answers · Free to read

Function: a special relation

From daily life

A vending machine gives exactly one drink for each button you press. One input, one output, no confusion.

Simple definition

A relation from A to B is a function f : A → B if every element of A has exactly one image in B. We write y = f(x); x is the input (argument) and y the output (image).

Formula: f : A → B, y = f(x)

123abc

In the figure: 1 and 2 both go to a (allowed). Each input has exactly one arrow. Not every output needs an arrow.

Rule 1: Is it a function — Test: (i) every element of A is used, (ii) no element has two arrows.

  • Easy: {(1, 2), (2, 3), (3, 4)} from {1, 2, 3}: function.
  • Medium: {(1, 2), (1, 3), (2, 4)}: not a function, since 1 has two images.
  • Exam-type: {(1, 2), (2, 2), (3, 2)}: function (constant).

Rule 2: Value of a function — Replace x by the given number.

  • Easy: f(x) = 3x − 2: f(2) = 4.
  • Medium: f(x) = 3x − 2: f(−1) = -5.
  • Exam-type: f(x) = x² + 1: f(a + 1) = a² + 2a + 2.
Remember

Many inputs may share one output, but one input can never give two outputs.

Common mistake

Thinking a function needs different outputs for different inputs.

Memory clue

Function = One input, One output.

Spot the difference

RelationFunction
Any subset of A × BEach x has exactly one image
One x may have several yOne x has one y

Practice questions

Easy
  1. 1Is {(1, 5), (2, 5)} a function?
  2. 2Is {(1, 5), (1, 6)} a function?
  3. 3f(x) = 2x + 3. Find f(4).
Medium
  1. 4f(x) = x² − 4. Find f(−3).
  2. 5f(x) = 5x − 7. Find x if f(x) = 18.
  3. 6f(x) = x² + 2x. Find f(3) − f(1).
Challenging
  1. 7f(x) = 4x − 1. Find f(x + 1) − f(x).
  2. 8From A = {1, 2} to B = {a, b, c}, how many functions exist?
Show answers (try them yourself first)
  1. 1Yes.
  2. 2No — 1 has two images.
  3. 311
  4. 45
  5. 55x = 25, so x = 5
  6. 615 − 3 = 12
  7. 74(x + 1) − 1 − (4x − 1) = 4
  8. 8Each of 2 inputs picks 1 of 3 outputs: 3² = 9

Domain, codomain and range of a real function

From daily life

A calculator shows "Error" when you take the square root of a negative number or divide by 0. Those inputs are outside the domain.

Simple definition

For a real function, the domain is the set of all real x for which f(x) is a real number. The range is the set of all outputs. Three checks: no division by 0, no square root of a negative number, and the usual rules for other operations.

-2-1012345678►

In the figure: f(x) = √(x − 3): domain is [3, ∞).

Rule 1: Find the domain — Find the values of x that break the rules, then remove them.

  • Easy: f(x) = 1/(x − 2): x ≠ 2, domain R − {2}.
  • Medium: f(x) = √(x − 3): need x − 3 ≥ 0, domain [3, ∞).
  • Exam-type: f(x) = 1/√(5 − x): need 5 − x > 0, domain (−∞, 5).

Rule 2: Find the range — Think about what values the output can take.

  • Easy: f(x) = x²: output ≥ 0, range [0, ∞).
  • Medium: f(x) = 2x + 1: every real value is possible, range R.
  • Exam-type: f(x) = √x: range [0, ∞).
Remember

For a fraction, the denominator can never be 0; for an even root, the inside can never be negative.

Common mistake

Allowing 0 under a square root. √0 = 0 is fine; negative is not.

Memory clue

Domain = Dos and Don'ts for x.

Practice questions

Easy
  1. 1Find the domain of f(x) = 1/(x − 7).
  2. 2Find the domain of f(x) = √(x + 4).
  3. 3Find the range of f(x) = x² + 3.
Medium
  1. 4Find the domain of f(x) = 1/(x² − 9).
  2. 5Find the domain of f(x) = √(6 − 2x).
  3. 6Find the range of f(x) = |x| + 1.
Challenging
  1. 7Find the domain of f(x) = √(x − 1) + 1/(x − 4).
  2. 8f : {1, 2, 3, 4} → N is f(x) = 2x + 1. Find its range.
Show answers (try them yourself first)
  1. 1R − {7}
  2. 2[−4, ∞)
  3. 3[3, ∞)
  4. 4x ≠ 3 and x ≠ −3, so R − {−3, 3}
  5. 56 − 2x ≥ 0 gives x ≤ 3, so (−∞, 3]
  6. 6[1, ∞)
  7. 7x ≥ 1 and x ≠ 4, so [1, 4) ∪ (4, ∞)
  8. 8{3, 5, 7, 9}

Some standard functions

From daily life

A taxi fare that adds a flat charge, a parking fee that is a fixed amount, or a ticket price rounded down to complete years: each is a familiar function pattern.

Simple definition

Important real functions: identity f(x) = x; constant f(x) = c; polynomial (like x² − 3x + 1); rational p(x)/q(x) with q(x) ≠ 0; modulus |x| (x if x ≥ 0, −x if x < 0); signum (1, 0 or −1 according as x is positive, zero or negative); greatest integer [x] = the largest integer not greater than x.

Formula: |x| = x for x ≥ 0, |x| = −x for x < 0

Rule 1: Modulus — Distance from 0, never negative.

  • Easy: |5| = 5, |−5| = 5.
  • Medium: |3 − 8| = 5.
  • Exam-type: |x| = 4 gives x = 4 or x = −4.

Rule 2: Greatest integer [x] — Step down to the integer on the left (or the number itself).

  • Easy: [3.7] = 3.
  • Medium: [−2.3] = -3 (not −2).
  • Exam-type: [5] = 5 and [0.99] = 0.

Rule 3: Signum — Only the sign matters.

  • Easy: sgn(8) = 1.
  • Medium: sgn(−0.5) = −1.
  • Exam-type: sgn(0) = 0.
Remember

The graph of |x| is a V with its corner at the origin; the graph of [x] is a staircase.

Common mistake

Writing [−2.3] = −2. The greatest integer not exceeding −2.3 is −3.

Memory clue

[x] = floor: slide DOWN.

Spot the difference

x[x]
Always ≥ 0Can be negative
Range [0, ∞)Range Z (integers)

Practice questions

Easy
  1. 1Find |−9|.
  2. 2Find [6.4].
  3. 3Find [−0.5].
Medium
  1. 4Solve |x| = 7.
  2. 5Solve |x − 3| = 2.
  3. 6Find [2.5] + [−2.5].
Challenging
  1. 7What is the range of f(x) = [x]?
  2. 8Find f(−3) + f(0) + f(4) for f(x) = sgn(x).
Show answers (try them yourself first)
  1. 19
  2. 26
  3. 3-1
  4. 4x = 7 or x = −7
  5. 5x − 3 = 2 or −2, so x = 5 or 1
  6. 62 + (-3) = -1
  7. 7The set of all integers Z.
  8. 8−1 + 0 + 1 = 0

Algebra of real functions

From daily life

Total bill = food bill + service charge; profit = income − expense. We add or multiply two rules at the same x.

Simple definition

For real functions f and g with a common domain: (f + g)(x) = f(x) + g(x); (f − g)(x) = f(x) − g(x); (fg)(x) = f(x)·g(x); (f/g)(x) = f(x)/g(x), where g(x) ≠ 0; (cf)(x) = c·f(x).

Formula: (f + g)(x) = f(x) + g(x)

Rule 1: Add, subtract, multiply — Work at the same x and then simplify.

  • Easy: f(x) = 2x, g(x) = x + 3: (f + g)(x) = 3x + 3.
  • Medium: f(x) = x², g(x) = x − 1: (f − g)(x) = x² − x + 1.
  • Exam-type: f(x) = x, g(x) = x + 2: (fg)(x) = x² + 2x.

Rule 2: Quotient — Exclude the x for which g(x) = 0.

  • Easy: f(x) = x + 1, g(x) = x − 3: (f/g)(x) = (x + 1)/(x − 3), x ≠ 3.
  • Medium: f(x) = x², g(x) = x: (f/g)(x) = x, x ≠ 0.
  • Exam-type: Value check: (f/g)(5) = 6/2 = 3.
Remember

The domain of f + g, f − g and fg is the common part of the two domains; for f/g also remove the zeros of g.

Common mistake

Dividing f by g and forgetting x ≠ zeros of g.

Memory clue

Same x, then operate.

Practice questions

Easy
  1. 1f(x) = x + 4, g(x) = 2x. Find (f + g)(x).
  2. 2f(x) = 3x, g(x) = x − 2. Find (f − g)(x).
  3. 3f(x) = x + 1, g(x) = x − 1. Find (fg)(x).
Medium
  1. 4f(x) = x², g(x) = x + 5. Find (f + g)(2).
  2. 5f(x) = 2x + 3, g(x) = x − 1. Find (f/g)(3).
  3. 6Find the domain of f/g for f(x) = x, g(x) = x − 6.
Challenging
  1. 7f(x) = x², g(x) = 3x. Find x for which (f − g)(x) = 0.
  2. 8f(x) = 2x − 1, g(x) = 4x + 5. Find (3f − g)(x).
Show answers (try them yourself first)
  1. 13x + 4
  2. 22x + 2
  3. 3x² − 1
  4. 44 + 7 = 11
  5. 59/2
  6. 6All real x except 6.
  7. 7x² − 3x = 0, so x = 0 or 3
  8. 86x − 3 − 4x − 5 = 2x − 8