Compare the function y = tanx and its reciprocal. Identify two similarities and two differences between the functions.
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https://www.jiskha.com/questions/1829517/compare-the-function-y-tanx-and-its-reciprocal-identify-two-similarities-and-two
To compare the function y = tan(x) with its reciprocal, we need to understand the properties of both functions. Let's break it down step-by-step:
1. Understanding the function y = tan(x):
- The tangent function (y = tan(x)) is a trigonometric function that relates the value of y to the angle x.
- It is defined as the ratio of the sine function to the cosine function: tan(x) = sin(x)/cos(x).
- The domain of tan(x) is all real numbers except the values where cos(x) equals zero, which occur at x = (2n + 1)pi/2 radians, where n is an integer.
2. Understanding the function y = 1/tan(x):
- The reciprocal function (y = 1/tan(x)) is obtained by taking the inverse of tan(x).
- It can also be written as y = cot(x), where cot(x) = 1/tan(x).
- Similar to the tangent function, the domain of 1/tan(x) is all real numbers except the values where tan(x) equals zero, which occur at x = npi radians, where n is an integer.
Now, let's compare the two functions:
Similarities:
1. Periodicity: Both the tangent function and its reciprocal have a periodic nature. They repeat their patterns after specific intervals.
2. Asymptotes: Both functions have vertical asymptotes since they become undefined when their denominators (cos(x) for tan(x) and sin(x) for 1/tan(x)) approach zero.
Differences:
1. Range: The tangent function (y = tan(x)) has a range of all real numbers, while its reciprocal (y = 1/tan(x)) has a range of all real numbers except 0. This is because division by zero is undefined.
2. Behavior near asymptotes: The tangent function approaches positive or negative infinity as it approaches its vertical asymptotes, whereas its reciprocal approaches zero.
To summarize, the tangent function (y = tan(x)) and its reciprocal (y = 1/tan(x)) have similarities in periodicity and vertical asymptotes, while differences can be observed in their range and behavior near asymptotes.