Let \(\alpha, \beta\) and \(\gamma\) be three positive real numbers Let \(f ( x )=\alpha x ^5+\beta x ^3+\gamma x , x \in R\) and \(g: R \rightarrow R\) be such that \(g(f(x))=x\) for all \(x \in R\) If \(a_1, a_2, a_3, \ldots, a_n\) be in arithmetic progression with mean zero, then the value of \(f\left(g\left(\frac{1}{n} \displaystyle\sum_{i=1}^n f\left(a_i\right)\right)\right)\)is equal to :
- 0
- 3
- 9
- 27
The Correct Option is A
Solution and Explanation
The correct option is (A) : 0
Consider a case when α = β = 0 then
\(f(x) = yx\)
\(g(x)=\frac{x}{y}\)
\(\frac{1}{n}\sum{^{n}_{i=1}}f(a_i)⇒\frac{y}{n}(a_1+a_2+....+a_n)\)
\(=0\)
\(⇒f(g(0))⇒f(0)\)
\(⇒0\)
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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