To determine the feature of the function [tex]\( g(x) = -4 \log (x - 8) \)[/tex], we'll analyze how the function behaves as [tex]\( x \)[/tex] increases.
Consider the structure of the function [tex]\( g(x) = -4 \log (x - 8) \)[/tex]:
1. Logarithmic Transformation: [tex]\( \log (x - 8) \)[/tex]
- The logarithmic function [tex]\( \log(x - 8) \)[/tex] increases as [tex]\( x \)[/tex] increases, provided [tex]\( x > 8 \)[/tex], since the logarithm of a larger number is greater.
2. Negative Coefficient: Multiplying by -4
- When we multiply the increasing function [tex]\( \log (x - 8) \)[/tex] by a negative coefficient (-4), it reverses the direction of change. This means that as [tex]\( x \)[/tex] increases, [tex]\( -4 \log (x - 8) \)[/tex] decreases because multiplying by a negative number flips the growth to shrinkage.
3. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity:
- As [tex]\( x \)[/tex] gets very large (approaches positive infinity), [tex]\( x - 8 \)[/tex] also gets very large.
- Consequently, [tex]\( \log (x - 8) \)[/tex] becomes very large.
- Since we are multiplying by -4, [tex]\( -4 \log (x - 8) \)[/tex] becomes very large in the negative direction (decreasing without bound).
Thus, the feature of the function [tex]\( g(x) = -4 \log (x - 8) \)[/tex] is that the value of the function decreases as [tex]\( x \)[/tex] approaches positive infinity.
Therefore, the correct feature described for [tex]\( g(x) \)[/tex] is:
A. The value of the function decreases as [tex]\( x \)[/tex] approaches positive infinity.
