SATHEE: Functions 3 Question 10

Functions 3 Question 10

10. Let $f$ be a one-one function with domain $x, y, z$ and range $1, 2, 3$ . It is given that exactly one of the following statements is true and the remaining two are false $f (x) = 1, f (y) \neq 1, f (z) \neq 2$ determine $f^{- 1} (1)$ .

(1982, 2M)

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Answer:

Correct Answer: 10. (b)

Solution:

Given, $f : [0, \infty) \to [0, \infty)$

Here, domain is $[0, \infty)$ and codomain is $[0, \infty)$ . Thus, to check one-one

Since, $f (x) = \frac{x}{1 + x} \Rightarrow f^{'} (x) = \frac{1}{(1 + x)^{2}} > 0, \forall x \in [0, \infty)$

$∴ f (x)$ is increasing in its domain. Thus, $f (x)$ is one-one in its domain. To check onto (we find range)

$\begin{aligned} Again, y = f (x) = \frac{x}{1 + x} \\ \Rightarrow y + y x = x \\ \Rightarrow x = \frac{y}{1 - y} \Rightarrow \frac{y}{1 - y} \geq 0 \end{aligned}$

Since, $x \geq 0$ , therefore $0 \leq y < 1$

i.e. Range $\neq$ Codomain

$∴ f (x)$ is one-one but not onto.

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Functions 3 Question 10

10. Let f be a one-one function with domain x,y,z and range 1,2,3. It is given that exactly one of the following statements is true and the remaining two are false f(x)=1,f(y)≠1,f(z)≠2 determine f−1(1).

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10. Let $f$ be a one-one function with domain $x, y, z$ and range $1, 2, 3$ . It is given that exactly one of the following statements is true and the remaining two are false $f (x) = 1, f (y) \neq 1, f (z) \neq 2$ determine $f^{- 1} (1)$ .