Exercise 1.2 — Relations and Functions

1. 1.2.15 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Show that the function f: R* -> R* defined by f(x)=1/x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*?

   Hint. For the second part, ask which real numbers can actually be written as 1/x for a natural number x.

   One-one: f(x1)=f(x2) means 1/x1=1/x2, so x1=x2.

   Onto: for any y in R*, take x=1/y, which is a non-zero real number, and f(x)=1/(1/y)=y.

   With domain N instead: one-one still holds (same argument, restricted to N). But onto fails: for y=2, we would need x=1/2 in N, which is not a natural number, so no preimage exists in N.

   ✦ f is one-one and onto on R*->R*. With domain N, f is still one-one but is no longer onto.

2. 1.2.26 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Check the injectivity and surjectivity of the following functions: (i) f: N -> N given by f(x)=x^2 (ii) f: Z -> Z given by f(x)=x^2 (iii) f: R -> R given by f(x)=x^2 (iv) f: N -> N given by f(x)=x^3 (v) f: Z -> Z given by f(x)=x^3.

   Hint. Squares and cubes behave very differently on positive-only domains (N) versus domains that include negatives (Z, R).

   (i) N->N, x^2: injective, because for positive naturals x1^2=x2^2 forces x1=x2 (no sign ambiguity); not surjective, since a value like 2 is never a perfect square.

   (ii) Z->Z, x^2: not injective, since f(-1)=1=f(1) even though -1 is not 1; not surjective, because negative numbers can never be squares.

   (iii) R->R, x^2: not injective for the same sign-cancellation reason as (ii); not surjective, since negative numbers are never squares.

   (iv) N->N, x^3: injective, because the cube function is strictly increasing on positive naturals; not surjective, since a value like 2 is never a perfect cube.

   (v) Z->Z, x^3: injective, because the cube function is strictly increasing over all of Z (positive, negative, and zero); not surjective, since a value like 2 is never a perfect cube.

   ✦ (i) Injective, not surjective. (ii) Neither injective nor surjective. (iii) Neither injective nor surjective. (iv) Injective, not surjective. (v) Injective, not surjective.

3. 1.2.34 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Prove that the Greatest Integer Function f: R -> R, given by f(x)=[x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

   Hint. Find two different inputs with the same greatest-integer value, and a target value the function can never hit.

   Not one-one: f(1.5)=1 and f(1.2)=1, but 1.5 is not equal to 1.2.

   Not onto: [x] is always an integer, so a non-integer value such as 0.5 in the co-domain R is never the image of any x.

   ✦ f is neither one-one nor onto, confirmed by the counterexamples f(1.5)=f(1.2)=1 and the unreachable value 0.5.

4. 1.2.44 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Show that the Modulus Function f: R -> R, given by f(x)=|x|, is neither one-one nor onto, where |x| is x if x is positive or 0 and |x| is -x if x is negative.

   Hint. Modulus collapses a positive and a negative number to the same value, and never produces a negative output.

   Not one-one: f(-1)=1=f(1), but -1 is not equal to 1.

   Not onto: |x| is always >=0, so a negative value such as -1 in the co-domain R is never the image of any x.

   ✦ f is neither one-one nor onto.

5. 1.2.54 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Show that the Signum Function f: R -> R, given by f(x)=1 if x>0, 0 if x=0, -1 if x<0, is neither one-one nor onto.

   Hint. Every positive number gives the same output, and the range only has three possible values.

   Not one-one: f(2)=1=f(5), but 2 is not equal to 5.

   Not onto: the range of f is only {-1,0,1}; a value such as 0.5 in the co-domain R is never achieved.

   ✦ f is neither one-one nor onto.

6. 1.2.63 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Let A={1,2,3}, B={4,5,6,7} and let f={(1,4),(2,5),(3,6)} be a function from A to B. Show that f is one-one.

   Hint. List the images of every input and check they are all different.

   f(1)=4, f(2)=5, f(3)=6. All three inputs give distinct outputs, so no two different inputs share an image.

   ✦ f is one-one (note it is not onto, since 7 in B has no preimage, though the question only asks about one-one-ness).

7. 1.2.75 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f: R -> R defined by f(x)=3-4x. (ii) f: R -> R defined by f(x)=1+x^2.

   Hint. A linear function with nonzero slope is always bijective on R; a function bounded below can never be onto R.

   (i) One-one: f(x1)=f(x2) gives 3-4x1=3-4x2, so x1=x2. Onto: for any y, x=(3-y)/4 solves f(x)=y. So f is bijective.

   (ii) Not one-one: f(1)=2=f(-1), but 1 is not equal to -1. Not onto: since x^2>=0 always, 1+x^2>=1, so no x maps to a value like 0, which is in the co-domain R but below the function's actual range.

   ✦ (i) Bijective (both one-one and onto). (ii) Neither one-one nor onto.

8. 1.2.84 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Let A and B be sets. Show that f: A x B -> B x A such that f(a,b)=(b,a) is bijective function.

   Hint. Swapping the coordinates of an ordered pair is its own inverse operation.

   One-one: if f(a1,b1)=f(a2,b2), then (b1,a1)=(b2,a2), so b1=b2 and a1=a2, giving (a1,b1)=(a2,b2).

   Onto: given any (b,a) in B x A, the pair (a,b) in A x B satisfies f(a,b)=(b,a), so every element of B x A has a preimage.

   ✦ f is bijective.

9. 1.2.95 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

   Let f: N -> N be defined by f(n) = (n+1)/2 if n is odd, n/2 if n is even, for all n in N. State whether the function f is bijective. Justify your answer.

   Hint. Compute f at the first few natural numbers and look for a repeated output.

   f(1)=(1+1)/2=1 (n=1 is odd). f(2)=2/2=1 (n=2 is even). So f(1)=f(2)=1, but 1 is not equal to 2 — f is not one-one.

   (It is onto: every m in N is hit, by n=2m for instance, since f(2m)=2m/2=m — but failing one-one already rules out bijectivity.)

   ✦ f is not bijective, since it is not one-one (f(1)=f(2)=1).

10. 1.2.106 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

    Let A=R-{3} and B=R-{1}. Consider the function f: A -> B defined by f(x)=(x-2)/(x-3). Is f one-one and onto? Justify your answer.

    Hint. Cross-multiply to test one-one, and solve y=f(x) for x to test onto, checking the result stays inside A.

    One-one: f(x1)=f(x2) gives (x1-2)/(x1-3)=(x2-2)/(x2-3). Cross-multiplying and simplifying (as in the chapter's worked derivation) leads to x1=x2.

    Onto: solving y=(x-2)/(x-3) for x gives x=(3y-2)/(y-1), defined since y is not 1, and this x is never equal to 3 (substituting x=3 leads to a contradiction), so it always lies in A.

    ✦ Yes, f is both one-one and onto.

11. 1.2.112 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

    Let f: R -> R be defined as f(x)=x^4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

    Hint. Compare f at a positive and negative value, and check whether negative outputs are ever reached.

    f(-1)=1=f(1), so f is not one-one (many-one). The range of x^4 is [0,infinity), which is not all of R, so f is not onto.

    ✦ (D) f is neither one-one nor onto.

12. 1.2.122 marksNCERT Class 12 Mathematics, Relations and Functions, Reprint 2026-27

    Let f: R -> R be defined as f(x)=3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

    Hint. This is a straight line through the origin with nonzero slope.

    One-one: 3x1=3x2 implies x1=x2. Onto: for any y, x=y/3 satisfies f(x)=y, and y ranges over all of R.

    ✦ (A) f is one-one onto.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 12 Mathematics textbook, Reprint 2026-27 (lemh101.pdf) — Exercise 1.1 (16 questions), Exercise 1.2 (12 questions), plus the chapter's Miscellaneous Exercise (7 questions), 35 questions total. The old stub had no real exercise coverage at all (only an invented five-question 'Self-Test') and taught an entire 'binary operations' section that has no actual teaching content anywhere in the current book — the book's introduction and summary paragraphs still mention it, but the section itself, along with any example or exercise question on it, has been removed from this edition. Every proof-based answer in this file was independently re-derived from the definitions (not copied from a key), including the finite-set onto-functions count (n!) via the chapter's own one-one-iff-onto shortcut, and Miscellaneous Q5's function-equality check verified by direct computation of both f and g at all four points of the domain.. Questions are referenced from the NCERT textbook for identification.