We obtain the domain and range of each function using its graph. Also, the range of the function is the part of the vertical axis where the graph exists. From each given graph, the domain of the function corresponding to the graph is the part of the horizontal axis where the graph exists. Let us identify which one of the graphs represents a function whose domain is − ∞, 1 2 and whose range is [ 0, ∞ [. This leads to the range of □ ( □ ), which is [ 0, ∞ [. Since the range of □ ( □ ) = 1 − 2 □ is all real numbers, √ □ ( □ ) must have the same range as √ □. Recall that the range of the square root function √ □ is [ 0, ∞ [. Hence, the domain of □ ( □ ) is − ∞, 1 2 . Rearranging this inequality leads to □ ≤ 1 2, which is written as − ∞, 1 2 in interval notation. Hence, the domain of this function is the set of □-values such that We know that the domain of √ □ ( □ ) is the set of □-values satisfying Let us use the domain and the range of □ ( □ ) = √ 1 − 2 □ to identify the graph. Let us consider another example for obtaining the domain of a composite square root function. Hence, any number in the interval [ 0, ∞ [ is a possible function value of □ ( □ ) = √ − □. This means that the number − □ satisfies In other words, for any number □ in the interval [ 0, ∞ [, we can find some number □ that satisfies □ = √ □. We know that the range of the square root function √ □ is [ 0, ∞ [. The range of a function is the set of all possible function values. This leads to □ ≤ 0, which is ] − ∞, 0 ] in interval notation. Hence, the domain of the given function is found by setting the expression inside the square root to be greater than or equal to zero. We recall that the square root cannot take a negative number as an argument. Example 1: Finding the Domain of Root FunctionsĬonsider the function □ ( □ ) = √ − □.
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