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Determine whether the function is one-to-one. g(x)= √x

User Esboych
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Final answer:

The function g(x) = √x is one-to-one because every non-negative x-value produces a unique square root, and it passes the horizontal line test within the domain of non-negative real numbers.

Step-by-step explanation:

To determine whether the function g(x) = √x is one-to-one, we should check if every y-value in the function's range corresponds to exactly one x-value in its domain. For the square root function √x, assuming the principal square root (meaning, the non-negative root), the function is indeed one-to-one. This is because for each non-negative x-value, there is one and only one non-negative square root.

We can visualize this by considering if the graph of the function passes the horizontal line test: any horizontal line should intersect the graph at no more than one point. Since for √x, every non-negative x-value produces a unique square root, the horizontal line test is passed, and thus the function is one-to-one. In contrast, if we were to include negative x-values, we would find that the square root is not defined for negative numbers in real numbers, thus maintaining the one-to-one nature of the function within its domain of non-negative real numbers.

User Kelley Robinson
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