If f(x) = frac{alpha x}{x + 1}, x neq -1. The | vedclass

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

Question

(D) Given $f(x) = \frac{\alpha x}{x + 1}$.We need to find $\alpha$ such that $f(f(x)) = x$.First,calculate $f(f(x)) = \frac{\alpha f(x)}{f(x) + 1}$.Su

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(D) Given $f(x) = \frac{\alpha x}{x + 1}$.We need to find $\alpha$ such that $f(f(x)) = x$.First,calculate $f(f(x)) = \frac{\alpha f(x)}{f(x) + 1}$.Substitute $f(x) = \frac{\alpha x}{x + 1}$ into the expression:$f(f(x)) = \frac{\alpha \left( \frac{\alpha x}{x + 1} ight)}{\frac{\alpha x}{x + 1} + 1} = \frac{\frac{\alpha^2 x}{x + 1}}{\frac{\alpha x + x + 1}{x + 1}} = \frac{\alpha^2 x}{(\alpha + 1)x + 1}$.We are given $f(f(x)) = x$,so:$\frac{\alpha^2 x}{(\alpha + 1)x + 1} = x$.Assuming $x 
eq 0$,we have $\frac{\alpha^2}{(\alpha + 1)x + 1} = 1$.$\alpha^2 = (\alpha + 1)x + 1$.For this to hold for all $x$ in the domain,the coefficients of $x$ must be zero and the constant terms must be equal:$\alpha + 1 = 0 \implies \alpha = -1$.Checking the constant term: $\alpha^2 = 1$. If $\alpha = -1$,then $(-1)^2 = 1$,which is true.Thus,the value of $\alpha$ is $-1$.

If $f(x) = \frac{\alpha x}{x + 1}, x \neq -1$. Then,for what value of $\alpha$ is $f(f(x)) = x$?

Similar Questions

Let $f$ be a composite function of $x$ defined by $f(u) = \frac{1}{u^2 + u - 2}$ and $u(x) = \frac{1}{x - 1}$. Then the number of points $x$ where $f$ is discontinuous is

If $f : R \to R$ and $g : R \to R$ are defined as $f(x) = 2x - |x|$ and $g(x) = 2x + |x|$, then

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=|x|$ and $g(x)=[x-3]$ for $x \in R$, then $\{g(f(x)):-\frac{8}{5} < x < \frac{8}{5}\}$ is equal to

If $g(x) = x^2 + x - 2$ and $(g \circ f)(x) = 2x^2 - 5x + 2$, then $f(x)$ is equal to

Let $f(x) = \frac{x+1}{x-1}$ for all $x \neq 1$. Let $f^1(x) = f(x)$,$f^2(x) = f(f(x))$ and generally $f^n(x) = f(f^{n-1}(x))$ for $n > 1$. Let $P = f^1(2) \cdot f^2(3) \cdot f^3(4) \cdot f^4(5)$. Which of the following is a multiple of $P$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module