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Chapter 4: Problem 67

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

Write the equation for a cosecant function satisfying the given conditions. $$\text { period: } 3 \pi ; \text { range: }(-\infty,-2] \cup[2, \infty)$$

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