Let α, β and γ be three positive real numbers. Let f(x)=α x⁵+β x³+γ x, x R and g: R → R be such that g(f(x))=x for all x R. If a₁, a₂, a₃,, a n be in arithmetic progression with mean zero, then the value of f(g(1/n Σ i=1 ⁿ f(a i))) is equal to:

Sets, Relations and Functions: JEE Main 2022 Mathematics question

JEE Main 2022 · 28 Jul, Morning shift · Q3830

JEE Main 202228 Jul, Morning shiftMathematicsMediumSets, Relations and Functions168× since 2002

Let

\alpha, \beta

and

\gamma

be three positive real numbers. Let

f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}

and

g: \mathbf{R} \rightarrow \mathbf{R}

be such that

g(f(x))=x

for all

x \in \mathbf{R}

. If

\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}}

be in arithmetic progression with mean zero, then the value of

f\left(g\left(\frac{1}{\mathrm{n}} \sum\limits_{i=1}^{\mathrm{n}} f\left(\mathrm{a}_{i}\right)\right)\right)

is equal to :