Consider the following statements : Statement 1: The function f R → R such that f(x)=x^3 for all x ∈ R is one-one. Statement 2: f(a) f(b) for all a, b ∈ ℝ if the function f is one-one. Which one of the following is

Correct answer: A. Both the statements are true and Statement 2 is the correct explanation of Statement 1. aligned & f(x)=x^3 & f(x_1)=x_1^3 & f(x_2)=x_2^3 & if x_1=x_2 & then f(x_1)=f(x_2) aligned So it is one-one function Hence option (1) is correct.

NDA 2015

Exam: NDA
Year: 2015
Subject: Mathematics
Chapter: Functions

NDA 2015 · Mathematics · Functions

Consider the following statements : Statement 1: The function f R →...

Consider the following statements : Statement

1:

The function f R → R such that

\mathrm{f}(\mathrm{x})=\mathrm{x}^3

for all

\mathrm{x} \in \mathrm{R}

is one-one. Statement

2: f(a) \Rightarrow f(b)

for all

a, b \in \mathbb{R}

if the function

f

is one-one. Which one of the following is correct in respect of the above statements?

Options

ABoth the statements are true and Statement 2 is the correct explanation of Statement 1.
BBoth the statements are true and Statement 2 is not the correct explanation of Statement 1.
CStatement 1 is true but Statement 2 is false.
DStatement 1 is false but Statement 2 is true.

Correct answer

A. Both the statements are true and Statement 2 is the correct explanation of Statement 1.

Step-by-step solution

\begin{aligned} & \mathrm{f}(\mathrm{x})=\mathrm{x}^3 \\ & \mathrm{f}\left(\mathrm{x}_1\right)=\mathrm{x}_1^3 \\ & \mathrm{f}\left(\mathrm{x}_2\right)=\mathrm{x}_2^3 \\ & \text{ if } \mathrm{x}_1=\mathrm{x}_2 \\ & \text{ then } \mathrm{f}\left(\mathrm{x}_1\right)=\mathrm{f}\left(\mathrm{x}_2\right) \end{aligned}

So it is one-one function Hence option (1) is correct.

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