Let f : $\R → \R$ be a bijective function such that for all x ∈ R, f(x) = $\sum\limits^{\infin}_{n=1}a_nx^n$ and $f^{-1}(x)=\sum\limits_{n=1}^{\infin}b_nx^n$ , where f -1 is the inverse function of f. If a 1 = 2 and a 2 = 4, then b 1 is equal to _________.

Question

Let f : $\R → \R$  be a bijective function such that for all x ∈ R, f(x) = $\sum\limits^{\infin}_{n=1}a_nx^n$  and  $f^{-1}(x)=\sum\limits_{n=1}^{\infin}b_nx^n$ , where f -1 is the inverse function of f. If a 1 = 2 and a 2 = 4, then b 1 is equal to _________.

cdquestions Admin · Accepted Answer

Inverse Function Coefficient Calculation  Given $ f(x) = \sum_{n=1}^{\infty} a_n x^n $ and $ f^{-1}(x) = \sum_{n=1}^{\infty} b_n x^n $, we know that $ f(f^{-1}(x)) = x $. According to the composition of functions: $\sum_{n=1}^{\infty} a_n \left( \sum_{m=1}^{\infty} b_m x^m \right)^n = x$ By expanding the powers, we use the composition rule: $a_1 b_1 x + (a_1 b_2 + a_2 b_1) x^2 + \dots = x$ Comparing the coefficients for $ x $, we equate: $a_1 b_1 x = x$ Thus, $ a_1 b_1 = 1 $. Knowing that $ a_1 = 2 $, we solve: $2 b_1 = 1$ Hence, $ b_1 = 0.5 $. Since $ 0.5 $ lies in the range $ (0.49, 0.51) $, the solution is validated.

Let f : \(\R → \R\) be a bijective function such that for all x ∈ R, f(x) = \(\sum\limits^{\infin}_{n=1}a_nx^n\) and \(f^{-1}(x)=\sum\limits_{n=1}^{\infin}b_nx^n\), where f^-1 is the inverse function of f. If a₁ = 2 and a₂ = 4, then b₁ is equal to _________.

Correct Answer: 0.49 - 0.51

Solution and Explanation

Inverse Function Coefficient Calculation

Top Questions on Functions of One Real Variable

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Let f : \(\R → \R\) be a bijective function such that for all x ∈ R, f(x) = \(\sum\limits^{\infin}_{n=1}a_nx^n\) and \(f^{-1}(x)=\sum\limits_{n=1}^{\infin}b_nx^n\), where f-1 is the inverse function of f. If a1 = 2 and a2 = 4, then b1 is equal to _________.