Question:
Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying
$f(x+y)=f(x)+f(y)+f(x) f(y)$ and $f(x)=x g(x)$
for all $x, y \in R $. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?
Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying
$f(x+y)=f(x)+f(y)+f(x) f(y)$ and $f(x)=x g(x)$
for all $x, y \in R $. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?
$f(x+y)=f(x)+f(y)+f(x) f(y)$ and $f(x)=x g(x)$
for all $x, y \in R $. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?
Updated On: Apr 25, 2024
- $f$ is differentiable at every $x \in R$
- If $g(0)=1$, then $g$ is differentiable at every $x \in R$
- The derivative $f^{\prime}(1)$ is equal to $1$
- The derivative $f^{\prime}(0)$ is equal to $1$
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The Correct Option is A, B, D
Solution and Explanation
(A) $f$ is differentiable at every $x \in R$
(B) If $g(0)=1$, then $g$ is differentiable at every $x \in R$
(D) The derivative $f^{\prime}(0)$ is equal to $1$
(B) If $g(0)=1$, then $g$ is differentiable at every $x \in R$
(D) The derivative $f^{\prime}(0)$ is equal to $1$
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Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions