f(x) = r
1 + q1 −
p1 − x2 find domain and range
Domain: All real numbers, as there are no values of x that would result in a division by zero or the square root of a negative number.
Range: [1-r, ∞) since the smallest value of the expression is 1-r, and it increases as x increases without bound.
To find the domain and range of the function f(x) = r/(1 + q1 − p1 − x^2), we need to consider any restrictions on the possible values of x and analyze the output values.
Domain:
The domain of a function consists of all the possible input values. In this case, the only potential restriction on x is that the denominator, 1 + q1 − p1 − x^2, should not equal zero since division by zero is undefined. Therefore, we need to solve the equation 1 + q1 - p1 - x^2 ≠ 0 for x.
1 + q1 - p1 - x^2 ≠ 0
q1 - p1 - x^2 ≠ -1
x^2 ≠ q1 - p1 - 1
This equation tells us that x^2 must be less than q1 - p1 - 1. Since the square of a real number is always non-negative, we can say that q1 - p1 - 1 should be greater than or equal to zero.
q1 - p1 - 1 ≥ 0
q1 - p1 ≥ 1
Therefore, the domain of the function f(x) is all real numbers for which q1 - p1 ≥ 1.
Range:
The range of a function represents all possible output values. Since the only variable affecting the output is x in this function, we need to analyze the behavior of the function as x varies.
The function f(x) = r/(1 + q1 − p1 − x^2) is a rational function. In general, a rational function's range can be all real numbers except for where the function is undefined. In this case, the function is undefined when the denominator, 1 + q1 − p1 − x^2, is equal to zero, which we have already accounted for in the domain analysis.
Therefore, the range of the function f(x) is all real numbers.
In summary:
Domain: All real numbers for which q1 - p1 ≥ 1.
Range: All real numbers.