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June 2011 p13 q10
769
Let \(f : x \mapsto 3x - 4, \; x \in \mathbb{R}\).
Sketch in a single diagram the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), making clear the relationship between the graphs.
Solution
The function given is \(f(x) = 3x - 4\).
To find the inverse, solve for \(x\) in terms of \(y\):
\(y = 3x - 4\)
\(y + 4 = 3x\)
\(x = \frac{y + 4}{3}\)
Thus, \(f^{-1}(x) = \frac{x + 4}{3}\).
The graph of \(y = f(x) = 3x - 4\) is a straight line with a slope of 3 and a y-intercept of -4.
The graph of \(y = f^{-1}(x) = \frac{x + 4}{3}\) is a straight line with a slope of \(\frac{1}{3}\) and a y-intercept of \(\frac{4}{3}\).
The line \(y = x\) is the line of symmetry between the function and its inverse.
According to the mark scheme, \(y = f(x)\) is correct in the 1st and 4th quadrants, \(y = f^{-1}(x)\) is correct in the 1st and 2nd quadrants, and \(y = x\) is marked or quoted.
