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June 2004 p1 q10
772
Given the function \(g : x \mapsto 2x + 3\), where \(x \in \mathbb{R}\), sketch, in a single diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\), making clear the relationship between the graphs.
Solution
The function \(g(x) = 2x + 3\) is a linear function with a slope of 2 and a y-intercept of 3. To find the inverse function \(g^{-1}(x)\), we swap \(x\) and \(y\) and solve for \(y\):
\(x = 2y + 3\)
\(x - 3 = 2y\)
\(y = \frac{1}{2}(x - 3)\)
Thus, the inverse function is \(g^{-1}(x) = \frac{1}{2}(x - 3)\).
In the diagram, the graph of \(y = g(x)\) is a straight line with a slope of 2 and a y-intercept of 3. The graph of \(y = g^{-1}(x)\) is a straight line with a slope of \(\frac{1}{2}\) and a y-intercept of \(-\frac{3}{2}\).
The line \(y = x\) is also drawn to show the symmetry between \(y = g(x)\) and \(y = g^{-1}(x)\), as the inverse function is a reflection of the original function across the line \(y = x\).
