Exercise 2.3 — Relations and Functions

1. 2.3.13 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Which of the following are functions? Give reasons; if a function, give its domain and range. (i) {(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)}. (ii) {(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}. (iii) {(1,3),(1,5),(2,5)}.

   Hint. Check whether any first element repeats with a different second element -- that is the only way a relation can fail to be a function.

   (i) the six first elements 2,5,8,11,14,17 are all distinct, each pointing to a single image (1), so this is a function. (ii) the seven first elements are all distinct too, each with its own image, so this is also a function. (iii) the first element 1 appears twice, pointing to both 3 and 5, so this is NOT a function.

   ✦ Working through each part gives: (i) Function: domain={2,5,8,11,14,17}, range={1}. (ii) Function: domain={2,4,6,8,10,12,14}, range={1,2,3,4,5,6,7}. (iii) Not a function, since 1 has two different images.

2. 2.3.22 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Find the domain and range of: (i) f(x) = -x. (ii) f(x) = sqrt(9 - x^2).

   Hint. For (ii), the expression under the square root must be non-negative -- solve that inequality first to fix the domain.

   (i) -x is defined for every real x, and as x ranges over all reals so does -x. (ii) 9-x^2 >= 0 means x^2 <= 9, i.e. -3<=x<=3; over that domain, sqrt(9-x^2) ranges from 0 (at the endpoints) up to 3 (at x=0).

   ✦ Working through each part gives: (i) Domain = R, range = R. (ii) Domain = [-3, 3], range = [0, 3].

3. 2.3.32 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   A function f is defined by f(x) = 2x - 5. Find f(0), f(7), f(-3).

   Hint. Substitute each given value of x directly into the rule.

   f(0)=2(0)-5=-5. f(7)=2(7)-5=14-5=9. f(-3)=2(-3)-5=-6-5=-11.

   ✦ Working through each part gives: f(0)=-5, f(7)=9, f(-3)=-11.

4. 2.3.43 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   The function t maps Celsius to Fahrenheit by t(C) = 9C/5 + 32. Find (i) t(0) (ii) t(28) (iii) t(-10) (iv) the value of C when t(C)=212.

   Hint. For (iv), set the formula equal to 212 and solve the resulting linear equation for C.

   (i) t(0)=32. (ii) t(28)=9(28)/5+32=252/5+32=50.4+32=82.4. (iii) t(-10)=9(-10)/5+32=-18+32=14. (iv) 212=9C/5+32 gives 9C/5=180, so C=180x5/9=100 -- the boiling point of water, as expected.

   ✦ Working through each part gives: (i) 32 (ii) 82.4 (iii) 14 (iv) C = 100.

5. 2.3.53 marksNCERT Class 11 Mathematics, Relations and Functions, Reprint 2026-27

   Find the range of each function: (i) f(x)=2-3x, x in R, x>0. (ii) f(x)=x^2+2, x real. (iii) f(x)=x, x real.

   Hint. For (i), track what happens to the formula's value as x sweeps from just above 0 out to infinity.

   (i) as x increases from just above 0 toward infinity, 2-3x decreases from just below 2 toward negative infinity, so every value strictly less than 2 is achieved, but 2 itself is not (since x=0 is excluded). (ii) x^2 is always at least 0, so x^2+2 is always at least 2, and every value from 2 upward is achieved. (iii) x itself ranges over every real number.

   ✦ Working through each part gives: (i) Range = all reals less than 2, i.e. (-infinity, 2). (ii) Range = [2, infinity). (iii) Range = R.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh102.pdf) — three numbered exercises (2.1-2.3, 24 questions) plus the chapter's Miscellaneous Exercise (12 questions); Exercise 2.2 Q4's arrow-diagram figure (Fig 2.7) was rendered directly from the PDF at 250dpi and read visually to confirm the y=x-2 mapping, and the piecewise function boundaries in Miscellaneous Exercise Q1 were re-rendered at high resolution after the raw text extraction garbled the fraction/subscript layout, confirming f's pieces agree at x=3 (function) while g's pieces disagree at x=2 (not a function). Questions are referenced from the NCERT textbook for identification.