State the domain and range for each function.
y=2x^2/(x^2-1)
domain: ?
range: ?
domain x canot be +-1
range: all positive real numbers, including zero.
thank you.
range is all positive >2, as well as all negative
On the interval (-1,1) the numerator is positive and the denominator is negative.
To find the domain and range of a function, we need to consider any restrictions on the values of x and y.
In this case, we have the function:
y = 2x^2 / (x^2 - 1)
Let's start by determining the domain. The domain of a function is the set of all possible values for the independent variable (x) that will give a valid output (y). To find the domain, we need to consider any values of x that may cause the denominator (x^2 - 1) to equal zero.
In this function, the denominator (x^2 - 1) will be zero if x is equal to 1 or -1 (since (1^2 - 1) = 0 and (-1^2 - 1) = 0). Dividing by zero is undefined in mathematics, so these values are not included in the domain.
Therefore, the domain of the function y = 2x^2 / (x^2 - 1) is all real numbers except for x = 1 and x = -1.
Moving on to the range, the range of a function is the set of all possible values for the dependent variable (y) that can be obtained by plugging different values of x into the function. To determine the range, we need to analyze the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, both the numerator (2x^2) and the denominator (x^2 - 1) grow without bound. As a result, the function y becomes infinitely large in the positive direction.
Similarly, as x approaches negative infinity, both the numerator (2x^2) and the denominator (x^2 - 1) grow without bound. This causes the function y to become infinitely large in the negative direction.
Therefore, the range of the function y = 2x^2 / (x^2 - 1) is all real numbers except for zero (since the function never reaches the value of zero).
To summarize:
Domain: All real numbers except x = 1 and x = -1.
Range: All real numbers except zero.