Problem 60 - Algebra and Trigonometry

Question

Graph \(y=3^{x}\) and \(x=3^{y}\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of \(y=3^{x}\) is an increasing curve that passes through the point (0,1), and it approaches, but never reaches, the x-axis as \(x\) approaches negative infinity. The graph of \(x=3^{y}\) is a reflection of the graph of \(y=3^{x}\) along the line \(y=x\), and it approaches, but never touches, the positive y-axis as \(x\) approaches zero and \(y\) approaches negative infinity.

Step 1: Plot the graph of \(y=3^{x}\)

First, plot the graph of the function \(y=3^{x}\). An exponential function \(y=b^{x}\) where \(b>0\) and \(b≠1\) always has a horizontal asymptote at \(y=0\). For \(y=3^{x}\), when \(x\) is not a negative value, \(y\) will take a positive value, but as \(x\) becomes larger and larger, \(y\) will also become larger. When \(x\) is a negative value, \(y\) will take a small positive value but never reach zero. So the function graph increases as \(x\) moves to the right, and approaches \(y=0\) without ever touching it as \(x\) moves to the left.

Step 2: Plot the graph of \(x=3^{y}\)

Now plot the graph of the function \(x=3^{y}\). Since this function is the inverse of \(y=3^{x}\), to get the graph of \(x=3^{y}\), reflect the graph of \(y=3^{x}\) over the line \(y=x\). The graph of the inverse function will be symmetric to the graph of the original function with respect to the line \(y=x\). Hence, the graph of \(x=3^{y}\) increases as \(y\) moves up, and approaches \(x=0\) without ever touching it as \(y\) moves down.

Step 3: Combine both graphs in the same rectangular system

The final step is to draw both function graphs on the same coordinate grid. Plot several points for both functions and sketch the graphs. The graph of \(x=3^{y}\) should look like a mirror image of the graph of \(y=3^{x}\) reflected over the line \(y=x\).