What is the domain and range of y=5 tan x
The correct answer and explanation is:
The function given is y=5tan(x)y = 5 \tan(x), which is a transformation of the basic tangent function y=tan(x)y = \tan(x).
Domain:
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. The tangent function tan(x)\tan(x) is undefined wherever its argument equals an odd multiple of π2\frac{\pi}{2}, since the tangent function has vertical asymptotes at these points. Specifically, tan(x)\tan(x) is undefined at x=π2+nπx = \frac{\pi}{2} + n\pi, where nn is any integer.
Since the transformation y=5tan(x)y = 5 \tan(x) only affects the vertical stretching of the graph and does not change the points where the function is undefined, the domain of y=5tan(x)y = 5 \tan(x) remains the same as that of tan(x)\tan(x).
Thus, the domain of y=5tan(x)y = 5 \tan(x) is: x∈Rexceptx=π2+nπforn∈Zx \in \mathbb{R} \quad \text{except} \quad x = \frac{\pi}{2} + n\pi \quad \text{for} \quad n \in \mathbb{Z}
Range:
The range of a function refers to the set of all possible output values (y-values) that the function can take. The tangent function tan(x)\tan(x) has a range of all real numbers, i.e., (−∞,∞)(-\infty, \infty). The transformation y=5tan(x)y = 5 \tan(x) results in a vertical stretch by a factor of 5, meaning that for every output value of tan(x)\tan(x), the corresponding output value of y=5tan(x)y = 5 \tan(x) is 5 times greater.
Since the tangent function itself can take any real value, multiplying the output by 5 does not change the fact that the range of y=5tan(x)y = 5 \tan(x) is still all real numbers.
Thus, the range of y=5tan(x)y = 5 \tan(x) is: y∈Ry \in \mathbb{R}
Summary:
- Domain: x∈Rx \in \mathbb{R} except x=π2+nπx = \frac{\pi}{2} + n\pi, where nn is any integer.
- Range: y∈Ry \in \mathbb{R}.