what is the domain and range of the function
F(x)= 1/1-2cosx
domain is all reals except where the denominator is zero
1-2cosx = 0 when
cosx = 1/2
x = 2kπ ± π/3
so the domain is all reals except 2kπ ± π/3
To find the domain and range of the function f(x) = 1/(1 - 2cos(x)), we need to consider any restrictions on the possible values of x and the resulting values of f(x).
Domain:
The function will be defined for all real numbers x except when the denominator (1 - 2cos(x)) equals zero. As the cosine function oscillates between -1 and 1, it never reaches these extreme values simultaneously. Hence, there are no restrictions on the domain of this function, and the domain is all real numbers.
Domain: All real numbers (x ∈ ℝ)
Range:
To determine the range, we need to investigate the possible values of f(x) as x varies throughout the domain. Since the value of the numerator (1) is constant, it will not affect the range of the function. Thus, we only need to consider the range of the denominator.
The denominator, 1 - 2cos(x), will vary from a minimum of 1 - 2(-1) = 3 to a maximum of 1 - 2(1) = -1. Cosine can take on any value between -1 and 1, so the range of the denominator will be between 1 - 2(1) = -1 and 1 - 2(-1) = 3.
However, since the denominator cannot be zero, the range of the function will be all real numbers excluding zero.
Range: All real numbers except zero (f(x) ≠ 0, f(x) ∈ ℝ, f(x) ≠ 0)
To determine the domain and range of the function f(x) = 1 / (1 - 2cos(x)), we need to consider the restrictions and possible output values of the function.
Domain:
The domain of a function refers to the set of all possible input values. In this case, since we have a trigonometric function (cos(x)) in the denominator, we need to consider the values of x for which cos(x) does not equal zero, as division by zero is undefined.
The cosine function has a range between -1 and 1, inclusive. Therefore, the denominator of f(x) will never be zero since 1 - 2cos(x) will always be at least 1 + 2 = 3 or at most 1 - 2 = -1. Thus, there are no restrictions on the domain of the function f(x).
Therefore, the domain of f(x) = 1 / (1 - 2cos(x)) is all real numbers.
Range:
The range of a function refers to the set of all possible output values. To determine the range of this function, we need to analyze the behavior of the function for different values of x.
Since cos(x) ranges between -1 and 1, the value 2cos(x) can range between -2 and 2. Subtracting 2cos(x) from 1 will give us values between -1 and 3. Therefore, the function f(x) = 1 / (1 - 2cos(x)) will output values between -∞ and +∞.
In other words, the range of f(x) is all real numbers except 0.
To summarize:
Domain: All real numbers
Range: All real numbers except 0