Problem 70 - College Algebra

Question

Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The function \(f(x) = 2^{x}\) and its inverse \(f^{-1}(x) = \log_{2} x\) are symmetric to the line \(y=x\). They intersect at the point (1,1).

Step 1: Graph the function \(f(x) = 2^{x}\)

Choose suitable x-values and calculate the corresponding y-values. Make sure to choose negative, zero and positive x-values. For example, for \(x=-2, -1, 0, 1, 2\) we get the points \((-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4)\). Plot these points and draw a smooth curve through them which extends to the edges of the graph.

Step 2: Reflect the function in the line \(y=x\) to find its inverse

All points with coordinates \((x, y)\) on \(f\) will have coordinates \((y, x)\) on \(f^{-1}\). This means that the points from Step 1 become \((0.25, -2), (0.5, -1), (1, 0), (2, 1), (4, 2)\). Again, plot these points and draw a smooth curve through them.

Step 3: Verify the result

Make sure that both functions 'meet' at the point (1,1), because \(f(1)=f^{-1}(1)\). The curves of \(f\) and \(f^{-1}\) should be symmetric to the line \(y=x\).