For the function $f(x)=\sqrt{x-3}$,
a) State the domain of the function, using interval notation.
Domain of $f(x)$ : $\square$
b) Graph the function
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Draw:
c) Use the graph to determine the range of the function. Use interval notation.
Range of $f(x)$ : $\square$
Answer (2)
The domain is found by setting the expression inside the square root to be non-negative: x − 3 ≥ 0 , which gives x ≥ 3 .
The domain in interval notation is [ 3 , ∞ ) .
The range is all non-negative real numbers since the square root function always returns non-negative values.
The range in interval notation is [ 0 , ∞ ) .
The final answer is: Domain: [ 3 , ∞ ) , Range: [ 0 , ∞ ) .
Explanation
Understanding the Problem We are given the function f ( x ) = x − 3 . We need to find the domain and range of this function and also graph it.
Finding the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the square root function, the expression inside the square root must be non-negative. Therefore, we must have x − 3 ≥ 0 .
Expressing the Domain To find the domain, we solve the inequality x − 3 ≥ 0 . Adding 3 to both sides, we get x ≥ 3 . In interval notation, this is [ 3 , ∞ ) .
Finding the Range The range of a function is the set of all possible output values (y-values) that the function can produce. Since the square root function always returns non-negative values, and x can take any value greater or equal to 3, the smallest value of f ( x ) is f ( 3 ) = 3 − 3 = 0 = 0 . As x increases, f ( x ) also increases. Therefore, the range of the function is all non-negative real numbers, which is [ 0 , ∞ ) in interval notation.
Graphing the Function To graph the function, we can plot some points. We know that the function starts at x = 3 , where f ( 3 ) = 0 . When x = 4 , f ( 4 ) = 4 − 3 = 1 = 1 . When x = 7 , f ( 7 ) = 7 − 3 = 4 = 2 . The graph starts at the point ( 3 , 0 ) and increases as x increases.
Final Answer 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 we are modeling the distance a car travels as a function of time, the domain would be the set of all possible times (which cannot be negative), and the range would be the set of all possible distances (also non-negative). Similarly, in finance, if we are modeling the profit of a company as a function of the number of products sold, the domain would be the number of products sold (which cannot be negative), and the range would be the possible profit values. Knowing the domain and range helps us understand the limitations and possible values of the model.
The domain of the function f ( x ) = x − 3 is [ 3 , ∞ ) and the range is [ 0 , ∞ ) .
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