If \( f(x) = \frac{4x + 5}{6x - 4}, \, x \neq \frac{2}{3} \) and \( (fof)(x) = g(x) \), where \( g : \mathbb{R} - \left\{ \frac{2}{3} \right\} \rightarrow \mathbb{R} - \left\{ \frac{2}{3} \right\} \), then \( (gogogog)(4) \) is equal to
- \( -\frac{19}{20} \)
- \( \frac{19}{20} \)
- \( -4 \)
- 4
The Correct Option is D
Solution and Explanation
Given:
\[ f(x) = \frac{4x + 3}{6x - 4} \]
Compute \( g(x) \) as:
\[ g\left(\frac{4x + 3}{6x - 4}\right) = \frac{\left(\frac{4x + 3}{6x - 4}\right) + 3}{\left(\frac{4x + 3}{6x - 4}\right) - 4} = \frac{34x}{34} = x \]
Thus:
\[ g(x) = x \quad \implies \quad g(g(g(4))) = 4 \]
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