To analyze the relationship between the graphs of \( f(x) \) and \( g(x) \), we start from the given equation:
\[ f(x) = 4^x g(x) + f(x) + 5 \]
First, let's isolate \( g(x) \) in terms of \( f(x) \):
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Rearranging gives: \[ f(x) - f(x) = 4^x g(x) + 5 \] which simplifies to: \[ 0 = 4^x g(x) + 5 \] or \[ 4^x g(x) = -5 \]
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From this equation, we can express \( g(x) \) in terms of \( f(x) \): \[ g(x) = -\frac{5}{4^x} \]
Now we need to compare the graphs of \( f(x) \) and \( g(x) = -\frac{5}{4^x} \).
From the form of \( g(x) \):
- \( g(x) \) is a decreasing function that approaches 0 as \( x \to \infty \) and goes to \(-\infty\) as \( x \to -\infty\).
- \( g(x) \) has a horizontal asymptote at \( y=0 \).
On the other hand, since \( f(x) \) is defined in the context that it equals itself plus some additional terms and will contain characteristics of \( g(x) \) but with an additional offset and a scaling factor involving \( 4^x g(x) \), it suggests that the value of \( f(x) \) is shifted relative to \( g(x) \).
The equation's structure implies that the only transformation applied to \( g(x) \) resulting in \( f(x) \) is vertical, particularly since the equation eventually leads to a constant additional term.
Thus, the correct statement is: The graph of \( g(x) \) is shifted 5 units down from the graph of \( f(x) \).
Since this option isn't available in your list, we can consider the first option: The graph of \( f(x) \) is shifted up 5 units from the graph of \( g(x) \).
That's the conclusion that best fits the transformation inference from the relation \( f(x) = 4^x g(x) + f(x) + 5 \). The added complexity implies that if \( f \) is constantly offset vertically by +5, then it must appear in this context as being shifted above \( g \) by 5 units.
Hence, the statement that is true is: The graph of \( f(x) \) is shifted up 5 units from the graph of \( g(x) \).