f(x)=2x/√(1+9x²) Composed 10 Times, Find √(3α+1)

JEE Main 2024. Previous Year Question.

Problem

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \dfrac{2x}{\sqrt{1+9x^2}}$. If the composition of $f$ ten times, $(f \circ f \circ f \circ \cdots \circ f)(x) = \dfrac{2^{10}x}{\sqrt{1+9\alpha x^2}}$, then the value of $\sqrt{3\alpha+1}$ is equal to?

Key insight. Composing this function repeatedly doesn’t need to be done ten times by brute force. Computing just the first two or three compositions reveals a clean pattern in how the coefficient under the square root grows — and that pattern can be guessed, then confirmed, in closed form.

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Approach

Compute $f(f(x))$ and $f(f(f(x)))$ explicitly to see how the coefficient inside the square root changes with each composition. Once the pattern for the $n$-th composition is clear, plug in $n=10$ directly rather than composing ten times by hand.

Solution

Step 1 — Compute the second composition

$$f(f(x)) = \frac{2f(x)}{\sqrt{1+9f(x)^2}} = \frac{2\left(\dfrac{2x}{\sqrt{1+9x^2}}\right)}{\sqrt{1+9 \cdot \dfrac{4x^2}{1+9x^2}}} = \frac{4x}{\sqrt{1+45x^2}}$$

Step 2 — Compute the third composition and spot the pattern

Repeating the same substitution:

$$f(f(f(x))) = \frac{8x}{\sqrt{1+189x^2}}$$

Writing the coefficient inside each square root as $9 \times a_n$ (matching the problem’s form $1+9\alpha x^2$):

$$a_1 = 1, \quad a_2 = \frac{45}{9} = 5, \quad a_3 = \frac{189}{9} = 21$$

These match $a_n = \dfrac{4^n – 1}{3}$: check $a_1 = \tfrac{3}{3}=1$, $a_2=\tfrac{15}{3}=5$, $a_3=\tfrac{63}{3}=21$ — all confirmed.

Step 3 — Write the general $n$-th composition

$$\underbrace{f \circ f \circ \cdots \circ f}_{n \text{ times}}(x) = \frac{2^n x}{\sqrt{1+3(4^n-1)x^2}}$$

Step 4 — Match against $n=10$

Comparing $1 + 3(4^{10}-1)x^2$ with the given form $1+9\alpha x^2$:

$$9\alpha = 3(4^{10}-1) \implies \alpha = \frac{4^{10}-1}{3}$$

Step 5 — Compute $\sqrt{3\alpha+1}$

$$3\alpha + 1 = 3 \cdot \frac{4^{10}-1}{3} + 1 = (4^{10}-1) + 1 = 4^{10}$$

$$\sqrt{3\alpha+1} = \sqrt{4^{10}} = 4^5 = 1024$$

Answer

Common mistakes

• Trying to compose the function symbolically ten separate times. Two or three compositions are enough to reveal the pattern in the coefficient — extrapolating is far faster and less error-prone than grinding through all ten.
• Losing track of the factor of 3 vs. 9 between $a_n$ and the coefficient $9\alpha$ in the problem’s given form. Writing the coefficient explicitly as $9 \times a_n$ from the start avoids this mismatch.

Practise next

• For $g(x) = \dfrac{3x}{\sqrt{1+4x^2}}$, find the coefficient $\beta$ such that composing $g$ with itself 5 times gives $\dfrac{3^5x}{\sqrt{1+4\beta x^2}}$, using the same pattern-spotting method.