Board 2024 Class 12 Maths Q1 Solution: Is f(x) = 4x + 3 on R+ one-one or onto?

The question

A function $f : R_+ \to R$ (where $R_+$ is the set of all non-negative real numbers) defined by $f(x) = 4x + 3$ is :

(A) one-one but not onto

(B) onto but not one-one

(C) both one-one and onto

(D) neither one-one nor onto

Exam answer

Let $x_1, x_2 \in R_+$ ⁠.

\begin{aligned} f(x_1) &= f(x_2) \\ \Rightarrow 4x_1 + 3 &= 4x_2 + 3 \\ \Rightarrow 4x_1 &= 4x_2 \\ \Rightarrow x_1 &= x_2 \end{aligned}

Why?

To prove one-one, start from $f(x_1) = f(x_2)$ and show that it forces $x_1 = x_2$ ⁠. Then two different inputs can never give the same output.

Here we subtract $3$ from both sides, then divide by $4$ ⁠.

So $f$ is one-one.

For $x \ge 0$ ⁠, $f(x) = 4x + 3 \ge 3$ ⁠.

Take $0 \in R$ ⁠. Then $4x + 3 = 0$ gives $x = -\frac{3}{4} \notin R_+$ ⁠.

Why?

Onto means every element of the codomain $R$ is the image of some $x$ in the domain $R_+$ ⁠.

Why $f(x) \ge 3$ ⁠: start from $x \ge 0$ and do the same to both sides.

\begin{aligned} x &\ge 0 \\ 4x &\ge 0 \quad (\text{times } 4) \\ 4x + 3 &\ge 3 \quad (\text{add } 3) \end{aligned}

Solving $4x + 3 = 0$ one move at a time:

\begin{aligned} 4x + 3 &= 0 \\ 4x &= -3 \quad (\text{subtract } 3) \\ x &= -\frac{3}{4} \quad (\text{divide by } 4) \end{aligned}

The smallest value is $f(0) = 3$ ⁠, so the range is $[3, \infty)$ ⁠. A number below $3$ ⁠, such as $0$ ⁠, is never reached: the only $x$ that would give it is negative, and negative numbers are not in the domain.

Watch out: the codomain is $R$ ⁠, not $R_+$ ⁠. The same rule from $R$ to $R$ would be onto.

So $0$ has no pre-image in $R_+$ ⁠, and $f$ is not onto.

One-one but not onto rules out (B), (C) and (D).

Answer: (A) one-one but not onto

Where this comes from

Chapter 1, Relations and Functions: One-one and onto functions.

Question type: Prove one-one and onto

How to spot it: click to see

Type 5. Prove one-one and onto: the bijective function

Spot it: "show that $f$ is one-one and onto", or "is $f$ bijective?", with a formula like $f(x) = 3 - 4x$ ⁠.

Here: $4x_1 + 3 = 4x_2 + 3$ gives $x_1 = x_2$ ⁠, so $f$ is one-one. For onto, $4x + 3 = 0$ needs $x = -\frac34$ ⁠, which fails the domain check, as $R_+$ has no negative numbers.

Quick lesson · Worked NCERT solutions · Interactive lesson

Found an error?

Tell us if something on this page is wrong or unclear. Send feedback on this question →

Q1: Is f(x)=4x+3f(x) = 4x + 3f(x)=4x+3 on R+R_+R+​ one-one or onto?

Q1: Is $f(x) = 4x + 3$ on $R_+$ one-one or onto?