The function \(\rm f(x)=\frac{x}{1+x^2}\) from R to R is:

Question

Question

This question was previously asked in

AAI ATC Junior Executive 21 Feb 2023 Shift 1 Official Paper

Answer 1

Given:

\(\rm f(x)=\frac{x}{1+x^2}\) from R to R

Concept:

For one-one

If f(x1) = f(x2)

⇒ x₁ = x₂

Calculation:

For one-one

Let f(x₁) = f(x₂)

\(\rm \implies \frac{x_1}{x_1^2+1}=\frac{x_2}{x_2^2+1}\)

\(\rm \implies x_1x_2(x_2-x_1)=x_2-x_1\)

x₁ - x₂ = 0 or x₁x₂ = 1

So x1 = x2 or x1x2 = 1

Then, f is not one one function.

For onto

Let f(x) = y

\(\rm \implies \frac{x}{1+x^2}=y\)

For finding the inverse

swapping x & y

⇒ y/ 1 + y² = x

⇒ x + xy²= y

By solving the quadratic equation

y = (1 +- √1 - 4x²)/ 2x

y = 2 can't be attain Hence, For every value in codomain there is no such x exist in the domain of Real number

For y = 2 x should belongs to the Complex Number.

Hence, this function is not onto.

Hence the option (2) is correct.

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