Let α, β and γ be three positive real numbers. Let f(x) = αx⁵ + βx³ + γx, x ∈ R and g: R → R be such

Q: Wait, why do we care that the powers are odd? Can't we just solve it algebraically?

That is a fantastic question! While you *could* attempt to solve this using brute force, the JEE Advanced examiners are testing your ability to spot symmetry. Recognizing $f(x)$ as an odd function turns a potentially nightmarish calculation into a simple observation. In competitive exams, it is always about working smarter, not harder. Spotting the odd function property is the key that unlocks the entire problem.

Q: Does an Arithmetic Progression with mean zero *always* have symmetric terms?

Yes, absolutely! If the mean is zero, the sum of the terms must be zero. In an AP, the terms are distributed linearly: $a, a+d, a+2d, \dots$. If the sum is zero, the terms must be perfectly balanced around the center. For every positive term, there is a corresponding negative term. This symmetry is the engine that drives the cancellation in our sum.

Q: Why does $f(g(x)) = x$ hold true here?

Because the problem defines $g(f(x)) = x$. This is the formal definition of an inverse function. If $g$ is the inverse of $f$, then $g$ effectively 'undoes' the operation of $f$. Therefore, if you apply $f$ to the result of $g(x)$, you are essentially walking forward and then walking backward—you end up exactly where you started. It is a beautiful, fundamental property of inverse functions.

Question

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} 	o \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \dots, a_n$ be in arithmetic progression with mean zero, then the value of $f \left( g \left( \frac{1}{n} \sum_{i=1}^n f(a_i) ight) ight)$ is equal to :

Options:

(A) 0

(B) 3

(C) 9

(D) 27

Sigma Percentile · Accepted Answer

### The Beauty of Symmetry: Unlocking the Inverse

Welcome, future engineer! Today, we are going to dismantle a problem that looks like a monster but is actually a masterpiece of symmetry.

When you first look at $f(x) = \alpha x^5 + \beta x^3 + \gamma x$, you might feel the urge to start calculating derivatives or trying to find the inverse explicitly. Stop! Take a breath.

In JEE Advanced, the most complex-looking expressions are often hiding a secret path. Let's find it together.

### Phase 1: The Anatomy of the Function

Look at the function $f(x) = \alpha x^5 + \beta x^3 + \gamma x$. What do you see? The powers are $5, 3,$ and $1$.

Every single one of them is an odd number. In the world of functions, this is a massive signal. This is an **odd function**.

Mathematically, this means $f(-x) = -f(x)$. Geometrically, it means the graph is perfectly symmetric about the origin. If you rotate the graph $180$ degrees around the origin, it maps onto itself. This symmetry is not just a curiosity; it is our primary weapon.

### Phase 2: The Dance of the Arithmetic Progression

Now, let's look at the sequence $a_1, a_2, \dots, a_n$. We are told it is an Arithmetic Progression (AP) with a mean of zero.

The mean is defined as:

$$ \frac{1}{n} \sum_{i=1}^n a_i = 0 $$

If the mean is zero, then the sum of all terms must be zero: $\sum_{i=1}^n a_i = 0$. Imagine these terms on a number line.

Because it is an AP with a sum of zero, the terms are perfectly balanced around the origin. For every positive term $a_k$, there is a corresponding negative term $-a_k$. This is the 'symmetry' we were looking for!

### Phase 3: The Grand Collapse

Now, let's combine our two discoveries. We need to evaluate the sum $\sum_{i=1}^n f(a_i)$.

Because $f(x)$ is an odd function, we know that $f(-a_k) = -f(a_k)$. When we sum these up, the term $f(a_k)$ and the term $f(-a_k)$ will cancel each other out perfectly.

Since the entire AP is symmetric, every term has a partner that cancels it out. Therefore, the sum $\sum_{i=1}^n f(a_i)$ is exactly $0$.

### Phase 4: The Final Act

We are left with the expression $f(g(\frac{1}{n} \sum_{i=1}^n f(a_i)))$. We just proved that the sum is $0$.

So, the expression simplifies to $f(g(\frac{1}{n} \cdot 0))$, which is simply $f(g(0))$. Now, recall the definition of the inverse function given in the problem: $g(f(x)) = x$.

This implies that $g$ is the inverse of $f$. A fundamental property of inverse functions is that $f(g(x)) = x$.

By substituting $x=0$, we get $f(g(0)) = 0$. And just like that, the monster collapses into a simple **zero**.

You didn't need to solve for $\alpha, \beta,$ or $\gamma$. You didn't need to find the inverse function. You just needed to see the symmetry. Keep this mindset—look for the structure before you start the calculation!

Select Answer:

Visualized Solution

Analyze the Function $f (x)$

The Odd Function Property

Understanding the Inverse $g (x)$

Arithmetic Progression with Mean Zero

Symmetry of AP Terms

Summing the Function Values

Evaluating the Inner Mean

Final Conclusion

The Sigma Insight: Classification of Functions

The Beauty of Symmetry

Phase 1

Phase 2

Phase 3

Phase 4

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