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June 2002 p1 q10
774
Given the function \(f : x \mapsto 3x + 2\), \(x \in \mathbb{R}\), sketch, in a single diagram, the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), making clear the relationship between the two graphs.
Solution
The function \(f(x) = 3x + 2\) is a linear function with a slope of 3 and a y-intercept of 2. This means the graph of \(y = f(x)\) is a straight line that crosses the y-axis at (0, 2) and has a positive slope.
The inverse function \(f^{-1}(x)\) can be found by swapping \(x\) and \(y\) and solving for \(y\):
\(x = 3y + 2\)
\(x - 2 = 3y\)
\(y = \frac{x - 2}{3}\)
The graph of \(y = f^{-1}(x)\) is a line with a slope of \(\frac{1}{3}\) and an x-intercept of 2/3. This line is the reflection of \(y = f(x)\) in the line \(y = x\).
Thus, the graph of \(y = f(x)\) has a slope greater than 1 and a positive y-intercept, while the graph of \(y = f^{-1}(x)\) has a slope less than 1 and a positive x-intercept. The two graphs are reflections of each other across the line \(y = x\).
