The function \(f\) is such that \(f(x) = \frac{3}{2x+5}\) for \(x \in \mathbb{R}, x \neq -2.5\).
Obtain an expression for \(f^{-1}(x)\).
Solution
Let \(y = \frac{3}{2x+5}\).
Rearrange to make \(x\) the subject: \(2x + 5 = \frac{3}{y}\).
Multiply both sides by \(y\): \(y(2x + 5) = 3\).
Expand: \(2xy + 5y = 3\).
Rearrange to isolate \(x\): \(2xy = 3 - 5y\).
Divide by \(2y\): \(x = \frac{3 - 5y}{2y}\).
Thus, \(f^{-1}(x) = \frac{3 - 5x}{2x}\).
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