To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=√x? What is its domain?
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Step 1: The graph of the function ƒ(x)=√x is a curve that starts from the origin (0,0) and increases slowly as x increases. It is only present in the first quadrant of the Cartesian plane (where both x and y are positive or zero).
Step 2: The graph does not exist in the second quadrant (where x is negative and y is positive), the third quadrant (where both x and y are negative), or the fourth quadrant (where x is positive and y is negative). This is because the square root of a negative number is not a real number.
Step 3: Therefore, the graph of ƒ(x)=√x is the one that starts from the origin and increases slowly in the first quadrant only.
Step 4: The domain of a function is the set of all possible x-values. Since the square root of a negative number is not a real number, the domain of ƒ(x)=√x is all non-negative real numbers.
Step 5: In interval notation, the domain of ƒ(x)=√x is [0, ∞). This means that x can be any real number greater than or equal to 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Root Function
The square root function, denoted as ƒ(x) = √x, is a mathematical function that returns the non-negative square root of x. This function is defined only for non-negative values of x, meaning that it produces real number outputs only when x is greater than or equal to zero. The graph of this function starts at the origin (0,0) and increases gradually, forming a curve that approaches but never touches the x-axis.
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the square root function ƒ(x) = √x, the domain is restricted to non-negative real numbers, expressed as [0, ∞). This means that any negative input would result in an undefined output, as the square root of a negative number is not a real number.
Graphing functions involves plotting points on a coordinate plane to visually represent the relationship between the input (x) and output (ƒ(x)). For the square root function, the graph is a curve that starts at the origin and rises to the right, illustrating how the output increases as the input increases. Understanding how to interpret and sketch the graph of a function is essential for analyzing its behavior and properties.