How to find the domain and range of a radical function?
Understand the Problem
The question is asking how to determine the domain and range of a radical function. This involves identifying the set of possible input values (domain) and the corresponding set of output values (range) that the function can take.
Answer
The domain is \( [4, \infty) \) and the range is \( [0, \infty) \).
Answer for screen readers
The domain of the function ( f(x) = \sqrt{x - 4} ) is ( [4, \infty) ) and the range is ( [0, \infty) ).
Steps to Solve
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Identify the function
First, clarify the radical function you are working with. For example, let’s take the function ( f(x) = \sqrt{x - 4} ). -
Determine the domain
The domain of a radical function involves finding values of ( x ) for which the expression under the square root is non-negative. Set up the inequality:
$$ x - 4 \geq 0 $$
Solving this gives:
$$ x \geq 4 $$
Thus, the domain in interval notation is:
$$ [4, \infty) $$ -
Determine the range
Next, find the range of the function. The output values of ( f(x) ) depend on the minimum value of ( \sqrt{x - 4} ). Since the square root function produces non-negative outputs, we set:
$$ f(x) \geq 0 $$
The smallest value ( f(x) ) can be is 0, when ( x = 4 ). Thus, the range in interval notation is:
$$ [0, \infty) $$
The domain of the function ( f(x) = \sqrt{x - 4} ) is ( [4, \infty) ) and the range is ( [0, \infty) ).
More Information
When dealing with radical functions, understanding the restrictions imposed by the square root (or any radical) is crucial because it limits the input values. The domain represents all possible ( x )-values, while the range indicates the possible resulting ( f(x) )-values.
Tips
- Ignoring the inequality: Sometimes, students forget to set the expression under the square root to be non-negative, leading to incorrect domains.
- Misunderstanding range: Forgetting that the output of a square root function cannot be negative can result in an incorrect assessment of the range.
