What is the domain, range, zeros, symmetry, and is it even/odd of these functions? (how can you tell, i don't remember how you get all these, refresher please)
y= 1/x
y=[x]
y=sq rt. of x^2 -4
y=sin x
y=cos x
y=tan x
To determine the domain, range, zeros, symmetry, and whether the functions are even or odd, we need to analyze each function individually. Here is an explanation of how to find these characteristics for each function:
1. y = 1/x:
- Domain: The domain is the set of all real numbers except when x = 0 because dividing by zero is undefined. Therefore, the domain for this function is (-∞, 0) ∪ (0, +∞).
- Range: Since the reciprocal of a number can be any nonzero real number, the range for this function is (-∞, 0) ∪ (0, +∞).
- Zeros: The equation 1/x = 0 has no solutions since there is no value of x that will make the denominator equal to zero. Therefore, this function has no zeros.
- Symmetry: This function is neither symmetric about the x-axis nor the y-axis.
- Even/Odd: This function is neither even nor odd because f(-x) ≠ f(x) and f(-x) ≠ -f(x).
2. y = [x]:
- Domain: This function represents the greatest integer less than or equal to x. The domain for this function is all real numbers.
- Range: The range consists of all integer values since the output will always be an integer.
- Zeros: Zeros occur when the input gives an output of zero. In this case, the function [x] = 0 when x is an integer.
- Symmetry: This function is symmetric about the x-axis because the greatest integer function gives the same result for x and -x.
- Even/Odd: This function is even because f(-x) = f(x).
3. y = √(x^2 - 4):
- Domain: To determine the domain, we need to find the values of x that make the expression inside the square root greater than or equal to zero. Therefore, x^2 - 4 ≥ 0 becomes x ≤ -2 or x ≥ 2. So, the domain is (-∞, -2] ∪ [2, +∞).
- Range: The square root of x^2 - 4 can only be a non-negative number or zero. Hence, the range for this function is [0, +∞).
- Zeros: The function √(x^2 - 4) = 0 only when x = -2 or x = 2.
- Symmetry: This function is symmetric about the y-axis because replacing x with -x in the equation gives the same result.
- Even/Odd: This function is even because f(-x) = f(x).
4. y = sin(x):
- Domain: The domain for the sine function is all real numbers, so there are no restrictions.
- Range: The range for the sine function is -1 to 1, inclusive. Therefore, the range is [-1, 1].
- Zeros: The function sin(x) = 0 when x is a multiple of π (pi), i.e., x = 0, π, 2π, 3π, ...
- Symmetry: This function is symmetric about the origin (0,0) as sin(-x) = -sin(x).
- Even/Odd: This function is odd because f(-x) = -f(x).
5. y = cos(x):
- Domain: The domain for the cosine function is all real numbers, so there are no restrictions.
- Range: Similar to the sine function, the range for the cosine function is also -1 to 1, inclusive. Hence, the range is [-1, 1].
- Zeros: The function cos(x) = 0 when x is (n + 1/2)π, where n is an integer. For example, x = π/2, 3π/2, 5π/2, ...
- Symmetry: This function is symmetric about the y-axis because cos(-x) = cos(x).
- Even/Odd: This function is even because f(-x) = f(x).
6. y = tan(x):
- Domain: The domain for the tangent function depends on the values where cosine is not zero. Therefore, the domain for tan(x) is all real numbers except when cos(x) = 0. Hence, the domain is all real numbers excluding x = π/2, 3π/2, 5π/2, ...
- Range: The range for the tangent function is all real numbers.
- Zeros: The function tan(x) = 0 when x is an integer multiple of π, i.e., x = 0, π, 2π, 3π, ...
- Symmetry: This function is periodic with a period of π, and it has odd symmetry because tan(-x) = -tan(x).
- Even/Odd: This function is odd because f(-x) = -f(x).
Remember, these characteristics may vary depending on the specific restrictions or conditions imposed on the functions. However, these explanations provide a general understanding of how to determine the domain, range, zeros, symmetry, and whether the functions are even or odd.