To compare the functions \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt[3]{x} \), let's analyze each function in terms of their domains, ranges, and characteristics.
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Domains:
- \( f(x) = \sqrt{x} \) has a domain of \( [0, \infty) \) since the square root function is defined only for non-negative numbers.
- \( g(x) = \sqrt[3]{x} \) has a domain of \( (-\infty, \infty) \) because the cube root function is defined for all real numbers.
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Intersections:
- The functions intersect at the points where \( f(x) = g(x) \).
- For the points you provided:
- At \( (0, 0) \), both functions yield \( 0 \).
- At \( (1, 1) \), both functions yield \( 1 \).
- So, they do indeed intersect at \( (0, 0) \) and \( (1, 1) \).
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Increasing/Decreasing:
- \( f(x) = \sqrt{x} \) is an increasing function for \( x \geq 0 \).
- \( g(x) = \sqrt[3]{x} \) is also an increasing function for all \( x \).
Now, looking at the statements:
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"Both functions have a domain of \( (-\infty, \infty) \)" - This statement is false. \( f(x) = \sqrt{x} \) has a domain of \( [0, \infty) \), not \( (-\infty, \infty) \).
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"The graphs intersect at \( (0, 0) \) and \( (1, 1) \)" - This statement is true based on our analysis.
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"Both functions are increasing" - This statement is also true as both functions are increasing in their respective domains.
Thus, the false statement is:
"Both functions have a domain of (−∞,∞)".