Find the domain and range of the function
[tex]f(x)=\sqrt{x-3}[/tex]
Answer (1)
Determine the domain by ensuring the expression inside the square root is non-negative: x − 3 ≥ 0 , which gives x ≥ 3 .
Express the domain in interval notation: [ 3 , ∞ ) .
Find the minimum value of the function at x = 3 : f ( 3 ) = 0 .
Determine the range as all non-negative real numbers: [ 0 , ∞ ) . The final answer is: Domain: [ 3 , ∞ ) , Range: [ 0 , ∞ )
Explanation
Understanding the Problem We are given the function f ( x ) = s q r t x − 3 and we want to find its domain and range. The domain is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (f(x)-values) that the function can produce.
Finding the Domain For the function to be defined, the expression inside the square root must be greater than or equal to zero. This is because the square root of a negative number is not a real number. Therefore, we must have: x − 3 g e 0
Solving for x To solve the inequality x − 3 ≥ 0 , we add 3 to both sides: x g e 3
This means the domain of the function is all real numbers x such that x is greater than or equal to 3. In interval notation, the domain is [ 3 , ∞ ) .
Finding the Range Now let's find the range. Since the square root function always returns non-negative values, the smallest possible value of f ( x ) occurs when x is at its minimum value in the domain, which is x = 3 . In this case, f ( 3 ) = 3 − 3 = 0 = 0 .
Determining the Range As x increases beyond 3, the value of x − 3 also increases, and so does x − 3 . There is no upper bound on how large x can be, so there is no upper bound on the value of f ( x ) . Therefore, the range of the function is all non-negative real numbers. In interval notation, the range is [ 0 , ∞ ) .
Final Answer In summary, the domain of the function f ( x ) = x − 3 is [ 3 , ∞ ) , and the range is [ 0 , ∞ ) .
Examples
Understanding the domain and range of functions is crucial in many real-world applications. For example, if you're designing a solar panel system, you need to know the minimum amount of sunlight (input) required to generate a certain amount of electricity (output). The domain would represent the feasible sunlight hours, and the range would represent the possible electricity production levels. Similarly, in finance, when modeling investment returns, the domain could represent the initial investment amounts, and the range would represent the potential profit or loss. Knowing these boundaries helps in making informed decisions and predictions.
