Problem 67 - Precalculus Essentials

Question

If you are given the equation of a tangent function, how do you identify an \(x\) -intercept?

Step-by-Step Solution

Answer

The x-intercepts of the tangent function occur at points \(x = n\pi\), where n is any integer. These are the points where the function equals to zero.

Step 1: Understanding the Characteristic of Tangent Function

Understand that the tangent function, \(tan(x)\), equals zero when its input x is equal to multiple of \(\pi\). This is because for these inputs the point on the unit circle lies on the x-axis, making the y-coordinate (the sinus of the angle) equal to zero. As tangent function equals the ratio of the y-coordinate to the x-coordinate, any point on the x-axis will therefore make the value of the tangent function to be 0.

Step 2: Finding the X-Intercepts

Since the tangent function equals zero at multiples of \(\pi\), we can say that any x-value that is a multiple of \(\pi\) is an x-intercept of the function. Therefore, the solution to the problem are the x-values \(x = n\pi\) where n is any integer.

Step 3: Results

There are an infinite number of x-intercepts for the tangent function, and these intercepts occur at multiples of \(\pi\). In other words, the x-intercepts of the tangent function are the points \(x = n\pi\) where n is any integer.