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June 2012 p12 q10
767
Given the function \(f : x \mapsto 2x + 5\) for \(x \in \mathbb{R}\), sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) on the same diagram, making clear the relationship between the two graphs.
Solution
The function \(f(x) = 2x + 5\) is a linear function with a positive gradient of 2 and a \(y\)-intercept of 5.
To find the inverse function \(f^{-1}(x)\), we swap \(x\) and \(y\) in the equation \(y = 2x + 5\) and solve for \(y\):
\(x = 2y + 5\)
\(x - 5 = 2y\)
\(y = \frac{x - 5}{2}\)
Thus, \(f^{-1}(x) = \frac{x - 5}{2}\), which is also a linear function with a positive gradient of \(\frac{1}{2}\) and a \(y\)-intercept of \(-\frac{5}{2}\).
The line \(y = x\) is the line of symmetry between \(y = f(x)\) and \(y = f^{-1}(x)\), indicating that the graphs are reflections of each other across this line.
