To find the interval where both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are positive, let us analyze each function individually.
### Function [tex]\( f(x) \)[/tex]
Given the characteristics of the function [tex]\( f(x) \)[/tex]:
- Logarithmic function with a vertical asymptote at [tex]\( x = 0 \)[/tex].
- [tex]\( x \)[/tex]-intercept at [tex]\( (4, 0) \)[/tex].
- Decreasing over the interval [tex]\( (0, \infty) \)[/tex].
An appropriate form for such a function could be [tex]\( f(x) = -\log_b(x) \)[/tex]. This type of function:
- Has a vertical asymptote at [tex]\( x = 0 \)[/tex] because logarithm functions are undefined at 0.
- Intersects the x-axis at [tex]\( x = 4 \)[/tex] since [tex]\( -\log_b(4) = 0 \)[/tex].
- Is positive for smaller [tex]\( x \)[/tex] values within the interval [tex]\( 0 < x < 4 \)[/tex].
### Function [tex]\( g(x) \)[/tex]
Given the specific form of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \log_2(x + 3) - 2 \][/tex]
To determine where [tex]\( g(x) \)[/tex] is positive:
[tex]\[ \log_2(x + 3) - 2 > 0 \][/tex]
[tex]\[ \log_2(x + 3) > 2 \][/tex]
[tex]\[ x + 3 > 2^2 \][/tex]
[tex]\[ x + 3 > 4 \][/tex]
[tex]\[ x > 1 \][/tex]
So, [tex]\( g(x) \)[/tex] is positive for [tex]\( x > 1 \)[/tex].
### Overlapping Interval
Now, we need to find where both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are positive. This is the intersection of the two intervals [tex]\( 0 < x < 4 \)[/tex] and [tex]\( x > 1 \)[/tex].
The intersection of these two intervals is:
[tex]\[ 1 < x < 4 \][/tex]
Therefore, the interval where both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are positive is:
[tex]\[ \boxed{(1, 4)} \][/tex]
This interval [tex]\( (1, 4) \)[/tex] ensures that both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are positive.
