JEE Mains Maths · Relations and Functions

Range, One-One and Onto

Finding the set of values a function takes, by bounding, by the discriminant or by monotonicity, and using it to decide whether a function is one-one, onto, both or neither.

All Relations and Functions notes JEE Mains Maths strategy

Why this matters

Twenty-six PYQs, split evenly. Half ask for a range; the other half ask whether a function is one-one and onto, and the onto half of that question is a range question in disguise. Two ideas cover the page.

Concept 1 of 2: Finding the range

Three tools cover almost everything. If the formula is built from a bounded piece such as

\sin x

|x|

(x-a)^2

, shift and scale that piece's range. For

y=\frac{p(x)}{q(x)}

with quadratics, rearrange into a quadratic in

x

and require its discriminant to be

\ge0

. If the function is monotonic on an interval, its range runs between the values at the ends.

Definition

Bounded piece: $\sin x\in[-1,1]$ , $a\sin x+b\cos x\in[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$ .
Rational: $y\,q(x)-p(x)=0$ must have a real root: discriminant $\ge0$ .
Monotonic on $[a,b]$ : range between $f(a)$ and $f(b)$ .

Rational function

y=\frac{p(x)}{q(x)}\ \Rightarrow\ \Delta_x\ge0

Worked example

Find the range of

\frac{1}{5-2\sin x}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 30 January 2023 · Q166Moderate

Example 1 · Relations and Functions · Range, One-One and Onto

The range of the function

f(x) =\sqrt{3 - x}+\sqrt{2 + x}

is:

When the x² term vanishes

In the discriminant method, the value of

y

that makes the

x^2

coefficient zero turns the equation linear. Check it separately: it may or may not be in the range.

Drill 12 more on finding the range

Concept 2 of 2: Deciding one-one and onto

A function is one-one if different inputs give different outputs. It is enough that the function is strictly increasing or strictly decreasing, for instance with a derivative of one sign. To show it is not one-one, find two inputs with the same output: an even function, or a rational function that tends to the same value at both ends and turns in between. A function is onto if its range equals the stated codomain, so find the range and compare.

Definition

One-one: $f(a)=f(b)\Rightarrow a=b$ . Strictly monotonic suffices.
Onto: range = codomain.
Neither: a bounded, turning function into $\mathbb R$ .
Onto depends on the codomain given; one-one depends on the domain.

The two tests

\text{one-one: } f(a)=f(b)\Rightarrow a=b;\qquad \text{onto: } f(A)=B

Worked example

f:\mathbb R\to\mathbb R

f(x)=x^3+x

, one-one and onto?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 24 Jan 2025 · Q127Moderate

Example 2 · Relations and Functions · Range, One-One and Onto

The function

f:( - \infty,\infty) \rightarrow ( - \infty,1)

, defined by

f(x) =\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}

is :

Read the codomain

The same formula can be onto one codomain and not another. A range of

(-1,1)

is not onto

(-\infty,1)

, even though every value is below 1.

Drill 12 more on deciding one-one and onto

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Finding the range
Rational function
$y=\frac{p(x)}{q(x)}\ \Rightarrow\ \Delta_x\ge0$
Deciding one-one and onto
The two tests
$\text{one-one: } f(a)=f(b)\Rightarrow a=b;\qquad \text{onto: } f(A)=B$

Watch out for (2)

When the x² term vanishes
→ Finding the range
Read the codomain
→ Deciding one-one and onto

Test yourself on Relations and Functions

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Relations and Functions chapter test Drill all 26 questions