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Question
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
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Solution
\[Let f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow \frac{2y - 3}{4} = x\]
\[ \Rightarrow 2y - 3 = 4x\]
\[ \Rightarrow 2y = 4x + 3\]
\[ \Rightarrow y = \frac{4x + 3}{2}\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{4x + 3}{2} [\text{ from}\left( 1 \right)]\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{4x + 3}{2}\]
\[ \therefore \left( fo f^{- 1} \right)\left( 1 \right) = f\left( \frac{4\left( 1 \right) + 3}{2} \right) = f\left( \frac{7}{2} \right) = \frac{2\left( \frac{7}{2} \right) - 3}{4} = \frac{7 - 3}{4} = \frac{4}{4} = 1\]
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