If f(x) satisfies the relation fleft( frac{5x | vedclass

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

Question

(B) The given functional equation is $f\left( \frac{5x - 3y}{5 - 3} \right) = \frac{5f(x) - 3f(y)}{5 - 3}$.This is a form of the Cauchy functional equ

Vedclass · Accepted Answer

$\pi$

Vedclass · Answer

(B) The given functional equation is $f\left( \frac{5x - 3y}{5 - 3} \right) = \frac{5f(x) - 3f(y)}{5 - 3}$.This is a form of the Cauchy functional equation,which implies that $f(x)$ is a linear function of the form $f(x) = ax + b$.Given $f(0) = 1$,we substitute $x = 0$ into $f(x) = ax + b$ to get $b = 1$.Given $f'(0) = 2$,we differentiate $f(x) = ax + 1$ to get $f'(x) = a$,so $a = 2$.Thus,the function is $f(x) = 2x + 1$.We need to find the period of $\sin(f(x)) = \sin(2x + 1)$.The period of $\sin(kx + c)$ is given by $\frac{2\pi}{|k|}$.Here,$k = 2$,so the period is $\frac{2\pi}{2} = \pi$.

If $f(x)$ satisfies the relation $f\left( \frac{5x - 3y}{2} \right) = \frac{5f(x) - 3f(y)}{2}$ for all $x, y \in R$,with $f(0) = 1$ and $f'(0) = 2$,then the period of $\sin(f(x))$ is:

Similar Questions

Let $f$ be a non-zero real-valued continuous function satisfying $f(x+y) = f(x) \cdot f(y)$ for all $x, y \in R$. If $f(2) = 9$,then $f(6)$ is equal to

Let $R$ be the set of all real numbers and let $f$ be a function from $R$ to $R$ such that $f(x) + (x + \frac{1}{2}) f(1 - x) = 1$,for all $x \in R$. Then $2 f(0) + 3 f(1)$ is equal to

Let $f: R \rightarrow R$ be such that $f$ is injective and $f(x)f(y) = f(x+y)$ for all $x, y \in R$. If $f(x), f(y),$ and $f(z)$ are in $GP$,then $x, y,$ and $z$ are in:

Let a function $f : R \rightarrow R$ be defined such that $3f(2x^2 - 3x + 5) + 2f(3x^2 - 2x + 4) = x^2 - 7x + 9$ for all $x \in R$. Then the value of $f(5)$ is:

Let $f(x + y) = f(x) + f(y)$ for all $x, y \in R.$ Then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module