Compare the function y = tanx and its reciprocal. Identify two similarities and two differences between the functions
https://www.wolframalpha.com/input/?i=graph+y+%3D+tanx+%2C+y+%3D+cotx
y = tanx is in blue
the reciprocal, which would be y = 1/tanx = cotx is in pink
To compare the function y = tan(x) and its reciprocal function, we can start by understanding what these functions represent and then analyze their similarities and differences.
1. What is y = tan(x)?
The tangent function (tan(x)) is a trigonometric function that represents the ratio of the length of the side opposite an angle in a right triangle to the length of the adjacent side. In mathematical terms, tan(x) = opposite/adjacent.
2. What is the reciprocal function?
The reciprocal function of a function f(x) is given by 1/f(x). Therefore, the reciprocal of y = tan(x) is y = 1/tan(x).
Similarities:
1. Periodicity: Both the tangent function and its reciprocal are periodic functions. This means they repeat their values after a specific interval. For the tangent function, the period is π (or 180 degrees), and for its reciprocal, the period is also π or 180 degrees.
2. Vertical asymptotes: Both functions have vertical asymptotes. The tangent function has vertical asymptotes at x = (2n + 1)π/2 (where n is an integer), while its reciprocal has vertical asymptotes at x = nπ (where n is an integer).
Differences:
1. Range: The range of the tangent function is (-∞, +∞), meaning it takes on all possible real values. On the other hand, the range of its reciprocal function is (-∞, 0) U (0, +∞), excluding 0, as the reciprocal of 0 is undefined.
2. Symmetry: While the tangent function is an odd function, meaning it is symmetric about the origin (y-axis), its reciprocal function is an even function. An even function is symmetric about the y-axis.
In conclusion, the tangent function (y = tan(x)) and its reciprocal (y = 1/tan(x)) have similarities in terms of periodicity and vertical asymptotes. They differ in terms of range and symmetry.