Give an example of a radical function f(x) whose domain is (3,∞) and whose range is (-∞,0]
Can i plz get help
sqrt (x-3) is real only to the right of x=3
so try
-1/sqrt(x-3)
Well, we know that √x has a domain of x>=0, so we could try
√(x-3)
but that has domain [3,∞) and we want to exclude 3. So, let's divide, since we cannot divide by zero:
y = -1/√(x-3)
Now 3 is excluded, and luckily, the range is now also correct. You can check at wolframalpha.com to see:
http://www.wolframalpha.com/input/?i=-1%2F%E2%88%9A%28x-3%29
scroll down a bit past the graph, and it lists the domain and range.
Of course! I'd be happy to help.
To find a radical function that satisfies the given conditions, we can start by considering the properties of radical functions. A radical function is characterized by having a radical expression, such as a square root (√) or cube root (∛), involving the independent variable (x).
In this case, we want the domain of the function to be (3, ∞), which means that x can take on any values greater than 3. To ensure this, we need to select a radical function that does not have any constraints on its domain.
The range of the function is given as (-∞, 0]. This means that the output values of the function can be any negative number or zero. To achieve this, we will focus on a square root function since it can cover all negative values.
A simple square root function that satisfies the range condition is f(x) = -√x. By using the negative sign in front of the square root, the function will output negative values. Since the square root of any number greater than or equal to zero is always greater than or equal to zero, this function will give us the desired range of (-∞, 0].
Now, let's verify that the function f(x) = -√x meets all the requirements:
1. Domain: Since x appears under the square root, it is necessary that x ≥ 0 for the function to be defined. However, the given domain is (3, ∞), which already satisfies this requirement. Therefore, the domain condition is met.
2. Range: We have f(x) = -√x. Since √x ≥ 0 for any nonnegative value of x, multiplying it by -1 will give us negative values: -√x < 0. Therefore, the range of the function is indeed (-∞, 0].
Hence, the radical function f(x) = -√x satisfies the given conditions with a domain of (3, ∞) and a range of (-∞, 0].