Problem 59 - Algebra and Trigonometry

Question

Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

Plot exponential function \(y=2^{x}\) by picking suitable \(x\) values. Then calculate corresponding y values using the relation \(y=2^{x}\). For the inverse function, rewrite the equation as \(y=\log_2{x}\) and pick suitable values for \(x\), then calculate \(y\) values using the relation \(y=\log_2{x}\). The resulting plot should illustrate that the two graphs are reflections of each other over the line \(y=x\).

Step 1: Understand and Plot the Exponential Function \(y=2^{x}\)

The function \(y=2^{x}\) is an exponential function. Begin by choosing values for \(x\) and calculating corresponding \(y\). Choose values that are easy to calculate such as -2, -1, 0, 1, and 2. Plot these points to create a smooth curve for the function \(y=2^{x}\).

Step 2: Understand the Inverse Function \(x=2^{y}\)

The equation \(x=2^{y}\) can be rewritten as \(y=\log_2{x}\), which is the inverse function of \(y=2^{x}\). Here, \(y\) can only take values when \(x\) is positive. So, select values of \(x\) ranging from 0.25, 1, 2, 4 onward, and for each \(x\), calculate the corresponding value of \(y\) using the expression \(y=\log_2{x}\).

Step 3: Plot the Inverse Function \(x=2^{y}\)

Plot the points calculated in Step 2 on the same rectangular coordinate system. You will find that \(x\) is always positive in this case.