To analyze the functions \( f(x) = \sqrt{4x} \) and \( g(x) = \sqrt{4x - 3} \), let's determine their domains, ranges, and whether they are increasing or decreasing.
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Domains:
- For \( f(x) = \sqrt{4x} \):
- The expression inside the square root must be non-negative: \( 4x \geq 0 \Rightarrow x \geq 0 \). Therefore, the domain of \( f \) is \( [0, \infty) \).
- For \( g(x) = \sqrt{4x - 3} \):
- Similarly, \( 4x - 3 \geq 0 \Rightarrow 4x \geq 3 \Rightarrow x \geq \frac{3}{4} \). Thus, the domain of \( g \) is \( \left[\frac{3}{4}, \infty\right) \).
Since the domains are different, this implies the option concerning the domains would be “The functions have different domains.”
- For \( f(x) = \sqrt{4x} \):
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Ranges:
- The range of \( f(x) = \sqrt{4x} \) when \( x \) is in \( [0, \infty) \) starts from \( 0 \) and goes to \( \infty \), so the range is \( [0, \infty) \).
- The range of \( g(x) = \sqrt{4x - 3} \) when \( x \) starts from \( \frac{3}{4} \) also starts from \( 0 \) and goes to \( \infty \), since at \( x = \frac{3}{4} \), \( g\left(\frac{3}{4}\right) = \sqrt{0} = 0 \) and as \( x \) increases, \( g(x) \) increases without bound. The range is also \( [0, \infty) \).
This means the functions have the same range.
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Increasing/Decreasing behavior:
- Both functions \( f(x) \) and \( g(x) \) are square root functions, which are increasing functions. As \( x \) increases, both \( f(x) \) and \( g(x) \) also increase.
After analyzing all the points above, we can summarize:
- Domains: Different
- Ranges: Same
- Behavior: Both functions are increasing
Now, let's match our analysis with the provided options:
- Option #1: The functions have the same domain, the same range, both functions are decreasing. (Incorrect)
- Option #2: The functions have the same domain, different ranges, both functions are decreasing. (Incorrect)
- Option #3: The functions have different domains, the same range, both functions are increasing. (Incorrect)
- Option #4: The functions have different domains, different ranges, both functions are increasing. (Incorrect)
As per the correct analysis, none of the options perfectly match. However, the closest option based on the domain difference and the same range should indicate that they are both increasing.
Therefore, the most relevant option based on the analysis that accounts for different domains and same ranges is Option #3 but its minor mistake states both functions are increasing.
The correct option to select based on the available choices and the analysis above would be Option #3.
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