Given below are two statements:
Statement I: The function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=\frac{x}{1+|x|}$ is one-one.
Statement II: The function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=\frac{x^{2}+4x-30}{x^{2}-8x+18}$ is many-one.
In the light of the above statements, choose the correct answer from the options given below :
- (1) Both Statement I and Statement II are false.
- (2) Both Statement I and Statement II are true.
- (3) Statement I is false but Statement II is true.
- (4) Statement I is true but Statement II is false.
Solution:
Statement I: $f(x) = \frac{x}{1+|x|}$
If $x \ge 0$, $f(x) = \frac{x}{1+x}$. $f'(x) = \frac{1}{(1+x)^2} > 0$.
If $x < 0$, $f(x) = \frac{x}{1-x}$. $f'(x) = \frac{1}{(1-x)^2} > 0$.
Since $f'(x) > 0$ for all $x$, $f(x)$ is strictly increasing.
Thus, $f(x)$ is One-One.
Statement I is True.
Statement II: $f(x) = \frac{x^2+4x-30}{x^2-8x+18}$
To check if it is Many-One, let’s calculate $f(x)$ at a specific point, say $x=0$.
$$f(0) = \frac{-30}{18} = -\frac{5}{3}$$
Now, check if there is any other value $x$ for which $f(x) = -\frac{5}{3}$.
$$\frac{x^2+4x-30}{x^2-8x+18} = -\frac{5}{3}$$
$$3(x^2+4x-30) = -5(x^2-8x+18)$$
$$3x^2 + 12x – 90 = -5x^2 + 40x – 90$$
$$8x^2 – 28x = 0$$
$$4x(2x – 7) = 0$$
$$x = 0 \text{ or } x = \frac{7}{2}$$
Since $f(0) = f(3.5) = -5/3$, the function maps two distinct elements to the same image.
Thus, $f(x)$ is Many-One.
Statement II is True.
Both Statement I and Statement II are True.
Ans. (2)