If f(x) and g(x) are two real-valued function | vedclass

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

Question

(D) Given $f(g(x+y)) = f(g(x)) + f(g(y))$ ... $(i)$ Let $h(x) = f(g(x))$. Then the equation becomes $h(x+y) = h(x) + h(y)$,which is Cauchy's functiona

Vedclass · Accepted Answer

$\phi$

Vedclass · Answer

(D) Given $f(g(x+y)) = f(g(x)) + f(g(y))$ ... $(i)$ Let $h(x) = f(g(x))$. Then the equation becomes $h(x+y) = h(x) + h(y)$,which is Cauchy's functional equation. The solution to this is $h(x) = cx$ for some constant $c$. Given $g(1) = 2$ and $f(2) = 1$,we have $h(1) = f(g(1)) = f(2) = 1$. Since $h(1) = c(1) = 1$,we get $c = 1$. Thus,$h(x) = f(g(x)) = x$. Since $f(g(x)) = x$,$f$ and $g$ are inverse functions of each other. Therefore,$g(f(x)) = x$ for all $x \in R$. The function $g(f(x)) = x$ is a polynomial function,which is continuous everywhere on the set of real numbers $R$. Hence,the set of points where $g(f(x))$ is discontinuous is the empty set,denoted by $\phi$.

If $f(x)$ and $g(x)$ are two real-valued functions such that $f(g(x+y)) = f(g(x)) + f(g(y))$,$g(1) = 2$,and $f(2) = 1$,then the function $g(f(x))$ is discontinuous on the set

Similar Questions

Suppose $f : R \rightarrow (0, \infty)$ be a differentiable function such that $5f(x + y) = f(x) \cdot f(y), \forall x, y \in R$. If $f(3) = 320$,then $\sum_{n=0}^5 f(n)$ is equal to:

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:

$A$ real valued function $y = f(x)$ satisfies the relation $f\left( x - \frac{4}{9} \right) + 2x \le \frac{9}{4}x^2 + \frac{8}{9} \le f\left( x + \frac{4}{9} \right) - 2x$. The value of $f''(2)$ is

Let $f : R - \{0, 1\} \rightarrow R$ be a function such that $f(x) + f\left(\frac{1}{1-x}\right) = 1 + x$. Then $f(2)$ is equal to:

Let $f: N \times N \rightarrow N$ be a function such that $f(1,1)=2$,$f(m+1, n)=f(m, n)+2(m+n)$,and $f(m, n+1)=f(m, n)+2(m+n-1)$ for all $m, n \in N$. Find the value of $f(2,2)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module