Q. Given below are two statements :

Statement I : The function f : ℝ → ℝ defined by f(x) = x / (1 + |x|) is one-one.

Statement II : The function f : ℝ → ℝ defined by f(x) = (x² + 4x − 30) / (x² − 8x + 18) is many-one.

In the light of the above statements, choose the correct answer from the options given below :

Correct Answer: Both Statement I and Statement II are true

Explanation

Statement I:

Given

f(x) = x / (1 + |x|)

For x ≥ 0,

f(x) = x / (1 + x)

This function is strictly increasing for x ≥ 0.

For x < 0,

f(x) = x / (1 − x)

This function is also strictly increasing for x < 0.

Also, the range of the function for x ≥ 0 is (0, 1) and for x < 0 is (−1, 0). There is no overlap in outputs.

Hence, distinct inputs give distinct outputs for all real x.

So, Statement I is true.

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Statement II:

Given

f(x) = (x² + 4x − 30) / (x² − 8x + 18)

Factorising,

Numerator = (x + 10)(x − 3) Denominator = (x − 3)(x − 6)

So for x ≠ 3,

f(x) = (x + 10)/(x − 6)

Now,

f(4) = (14)/(−2) = −7 f(8) = (18)/(2) = 9

Also,

f(1) = (11)/(−5) = −11/5 f(11) = (21)/(5) = 21/5

Different values of x can give the same output, hence the function is many-one.

So, Statement II is true.

Therefore, the correct answer is:

Both Statement I and Statement II are true

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