To compare the graphs of the functions \( f(x) = \sqrt{4x} \) and \( g(x) = \sqrt[3]{4x} \), let's analyze their domains, ranges, and behavior.
Domain:
- For \( f(x) = \sqrt{4x} \), the expression under the square root must be non-negative, so \( 4x \geq 0 \) which means \( x \geq 0 \). Thus, the domain of \( f(x) \) is \( [0, \infty) \).
- For \( g(x) = \sqrt[3]{4x} \), there are no restrictions on \( x \) because the cube root function is defined for all real numbers. Thus, the domain of \( g(x) \) is \( (-\infty, \infty) \).
Since the domains are different, we can conclude that the first row in options is not applicable.
Range:
- The range of \( f(x) = \sqrt{4x} \) is \( [0, \infty) \) as the function outputs only non-negative values.
- The range of \( g(x) = \sqrt[3]{4x} \) is all real numbers, \( (-\infty, \infty) \), since the cube root can take negative values for negative inputs.
Since the ranges are different, the second row in options is also not applicable.
Behavior:
- The function \( f(x) = \sqrt{4x} \) is increasing for \( x \geq 0 \).
- The function \( g(x) = \sqrt[3]{4x} \) is also increasing for all real \( x \).
Since both functions are increasing, the fourth row in options is not applicable.
After analyzing both functions:
- They have different domains (Option #3).
- They also have different ranges (Option #2).
- Both are increasing.
Thus, the true statement is described by Option #3: The functions have different domains; the functions have different ranges; both functions are increasing.
So the correct option is:
Option #3
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