Find the domain and range of each function.
f(x) = -sqrt{x}
h(x) = sqrt{x + 1}
Let P = (x, y) be a point on the graph of y = x^2 - 8.
A. Express the distance d from P to the point (0, -1) as a function of x.
B. What is d if x = 0?
C. What is d if x = -1?
you know that sqrt(n) is defined only for n >= 0
So, sqrt(x+1) is defined only for x+1 >= 0
Sketch the graph and you can easily determine the range.
For P(x), just use the usual distance formula
sqrt((x-0)^2 + (y+1)^2) = sqrt(x^2 + (x^2-8+1)^2)
Now just plug in your values.
To find the domain and range of each function, we need to consider the restrictions on the inputs (domain) and the resulting outputs (range) for the given functions.
1. Function f(x) = -√x:
Domain:
Since the square root of a negative number is undefined for real numbers, the domain of this function is all non-negative real numbers (x ≥ 0).
Range:
The range represents the set of all possible outputs for a given function. In this case, since we take the negative square root of x, the range of the function is all negative real numbers. (-∞, 0].
2. Function h(x) = √(x + 1):
Domain:
For this function, any real number can be used as an input, since adding 1 to any real number will always result in a real number. So, the domain of this function is all real numbers (-∞, ∞).
Range:
The range of the function represents the set of all possible outputs. Since we are only taking the square root of a positive or zero number, the range of this function is all non-negative real numbers [0, ∞).
Overall, the domain and range of the given functions can be summarized as follows:
f(x) = -√x:
Domain: x ≥ 0
Range: (-∞, 0]
h(x) = √(x + 1):
Domain: (-∞, ∞)
Range: [0, ∞)