If f( x + y )=f( x ) f( y ) and Σ limits_{ x =1}^{∞} f( x )=2, x , y ∈ N where N is set of all

English

Gujarati

Hindi

1.Relation and Function

hard

If $f( x + y )=f( x ) f( y )$ and $\sum \limits_{ x =1}^{\infty} f( x )=2, x , y \in N$ where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is

$\frac{1}{9}$

$\frac{4}{9}$

$\frac{1}{3}$

$\frac{2}{3}$

(JEE MAIN-2020)

Solution

$f(x+y)=f(x) \cdot f(y)$

$\sum_{x=1}^{\infty} f(x)=2 \quad$ where $x, y \in N$

$f(1)+f(2)+f(3)+\ldots . \infty=2 \ldots .(1)($ Given $)$

Now for $f(2)$ put $x=y=1$

$f(2)=f(1+1)=f(1) \cdot f(1)=(f(1))^{2}$

$f(3)=f(2+1)=f(2) \cdot f(1)=(f(1))^{3}$

Now put these values in equation

$f(1)+(f(1))^{2}+\left[f(1)^{2}+\ldots \infty=2\right]$

$\frac{f(1)}{1-f(1)}=2$

$\Rightarrow f(1)=\frac{2}{3}$

Now $f(2)=\left(\frac{2}{3}\right)^{2}$

$f(4)=\left(\frac{2}{3}\right)^{4}$

then the value of $\frac{f(4)}{f(2)}=\frac{\left(\frac{2}{3}\right)^{4}}{\left(\frac{2}{3}\right)^{2}}=\frac{4}{9}$

Standard 12

Mathematics

If $f$ is an even function defined on the interval $(-5, 5),$ then four real values of $x$ satisfying the equation $f(x) = f\left( {\frac{{x + 1}}{{x + 2}}} \right)$ are

hard

(IIT-1996)

Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
{\,{x^3} – {x^2} + 10x – 5\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \le 1\,\,\,\,\,\,\,\,\,\,\,\,}\\
{ – 2x + {{\log }_2}({b^2} – 2),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, > 1\,\,\,\,\,\,\,\,\,\,\,\,}
\end{array}} \right.$ the set of values of $b$ for which $f(x)$ has greatest value at $x = 1$ is given by

normal

The domain of the definition of the function $f\left( x \right) = \frac{1}{{4 – {x^2}}} + \log \,\left( {{x^3} – x} \right)$ is

hard

(JEE MAIN-2019)

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,

normal

(KVPY-2019)

Domain of the function $f(x) = \sqrt {2 – {{\sec }^{ – 1}}x} $ is

normal

If $f( x + y )=f( x ) f( y )$ and $\sum \limits_{ x =1}^{\infty} f( x )=2, x , y \in N$ where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is

Solution

Similar Questions

If $f$ is an even function defined on the interval $(-5, 5),$ then four real values of $x$ satisfying the equation $f(x) = f\left( {\frac{{x + 1}}{{x + 2}}} \right)$ are

The domain of the definition of the function $f\left( x \right) = \frac{1}{{4 – {x^2}}} + \log \,\left( {{x^3} – x} \right)$ is

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements. $I.$ $f$ is an odd function. $II.$ $f$ is an even function. $III$. $f$ is differentiable everywhere. Then,

Domain of the function $f(x) = \sqrt {2 – {{\sec }^{ – 1}}x} $ is

Start a Free Trial Now

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,