To determine which form of \( g(x) \) maintains the same domain as \( f(x) \) and preserves other properties, including having a single x-intercept, analyze each option carefully.
A. \( g(x) = 10f(x) \)
- In this case, \( g(x) \) is simply \( f(x) \) scaled by a factor of 10. Scaling a function does not change its x-intercepts because the places where \( f(x) = 0 \) remain the same. Since the domain of \( f \) is \( D \), the domain of \( g \) remains \( D \).
B. \( g(x) = f(x + 10) \)
- Here, \( g(x) \) is \( f(x) \) shifted 10 units to the left. Shifting the input of a function horizontally does not affect the domain, so the domain of \( g \) is still \( D \). The x-intercept would be shifted, but this does not affect the number of x-intercepts.
C. \( g(x) = f(x) + 10 \)
- Adding 10 to \( f(x) \) shifts the entire graph of \( f \) upwards by 10 units. While the domain remains unchanged (still \( D \)), this vertical shift does not affect the x-intercepts directly unless \( f(x) \) originally crossed the x-axis.
D. \( g(x) = f(x) - 10 \)
- Subtracting 10 from \( f(x) \) shifts the graph of \( f \) downwards by 10 units. This also does not affect the domain, keeping it \( D \), and, like option C, modifies the y-values without changing the place of the x-intercepts directly.
Considering that both \( f \) and \( g \) must have a single x-intercept, which implies a single solution to \( f(x)=0 \) and subsequently any similar form of \( g(x) \) must not alter this count in terms of shifting the x-intercepts only, we deduced:
Correct Answer:
If all conditions based on internal analysis are preserved, the direct scaling \( g(x) = 10f(x) \) (option A) does not compromise the domain or the x-intercept count of the function. Thus:
The correct answer is: A. [tex]\( g(x) = 10f(x) \)[/tex]
