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Nov 2003 p1 q10
773
Given the function \(f: x \mapsto 2x - 5\), \(x \in \mathbb{R}\), sketch, on a single diagram, the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), making clear the relationship between these two graphs.
Solution
The function \(f(x) = 2x - 5\) is a linear function with a slope of 2 and a y-intercept of -5. To find the inverse, solve for \(x\) in terms of \(y\):
\(y = 2x - 5\)
\(y + 5 = 2x\)
\(x = \frac{y + 5}{2}\)
Thus, the inverse function is \(f^{-1}(x) = \frac{x + 5}{2}\).
Sketch the graph of \(y = 2x - 5\), which is a straight line with a slope of 2 and y-intercept at -5.
Sketch the graph of \(y = \frac{x + 5}{2}\), which is a straight line with a slope of \(\frac{1}{2}\) and y-intercept at \(\frac{5}{2}\).
The graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) should be symmetric about the line \(y = x\).
