Let A = {x in R mid x text{ is not a positive | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $f(x) = \frac{2x}{x-1}$.To check for injectivity,we find the derivative: $f'(x) = \frac{(x-1)(2) - 2x(1)}{(x-1)^2} = \frac{-2}{(x-1)^2}$.Sin

Vedclass · Accepted Answer

Injective but not surjective.

Vedclass · Answer

(D) Given $f(x) = \frac{2x}{x-1}$.To check for injectivity,we find the derivative: $f'(x) = \frac{(x-1)(2) - 2x(1)}{(x-1)^2} = \frac{-2}{(x-1)^2}$.Since $f'(x) To check for surjectivity,let $f(x) = y$. Then $y = \frac{2x}{x-1} \Rightarrow yx - y = 2x \Rightarrow x(y-2) = y \Rightarrow x = \frac{y}{y-2}$.For $f$ to be surjective,for every $y \in R$,there must exist an $x \in A$ such that $f(x) = y$.If $y = 2$,then $x = \frac{2}{0}$,which is undefined. Thus,$2$ has no pre-image in $A$.Also,if $y = 4$,$x = \frac{4}{4-2} = 2$. However,$2$ is a positive integer,so $2 
otin A$.Since there exist elements in the codomain $R$ that do not have a pre-image in $A$,$f$ is not surjective.Therefore,$f$ is injective but not surjective.

Let $A = \{x \in R \mid x \text{ is not a positive integer}\}$. Let a function $f$ be defined as $f: A \rightarrow R$ such that $f(x) = \frac{2x}{x-1}$. Then $f$ is:

Similar Questions

Let $f : R \rightarrow R$ be defined as $f(x) = 3^{-|x|} - 3^x + \operatorname{sgn}(e^{-x}) + 2$ (where $\operatorname{sgn}(x)$ denotes the signum function of $x$). Then which one of the following is correct?

If $f(x) = |\sin x| + |\cos x|$ and $g(x) = [x]$,then what is the period of $h(x) = g(f(x))$,where $[.]$ denotes the Greatest Integer Function $(G.I.F.)$?

If a set $A$ has $m$ elements and the set $B$ has $n$ elements,then the number of injections from $A$ to $B$ is

On the set of integers $Z$,define $f: Z \rightarrow Z$ as $f(n) = \begin{cases} \frac{n}{2}, & n \text{ is even} \\ 0, & n \text{ is odd} \end{cases}$. Then $f$ is:

If $f : R \rightarrow R$ such that $f(x) = 5x - 3\cos x - 4\sin x$,then the function $f(x)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module