To determine the end behavior of the function [tex]\( w(x) = 5 \cdot (1.25)^x + 4 \)[/tex], we will analyze the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( \infty \)[/tex].
### As [tex]\( x \rightarrow -\infty \)[/tex]
1. Consider the term [tex]\( (1.25)^x \)[/tex]. When [tex]\( x \)[/tex] becomes a very large negative number, [tex]\( (1.25)^x \)[/tex] approaches 0.
2. This is because any base greater than 1 raised to a negative power produces a fraction less than 1, and as the exponent becomes more negative, this fraction gets closer and closer to 0.
So, as [tex]\( x \)[/tex] tends to [tex]\( -\infty \)[/tex]:
[tex]\[ (1.25)^x \rightarrow 0 \][/tex]
Now substituting this into the function:
[tex]\[ w(x) = 5 \cdot (1.25)^x + 4 \][/tex]
[tex]\[ w(x) \rightarrow 5 \cdot 0 + 4 \][/tex]
[tex]\[ w(x) \rightarrow 4 \][/tex]
### As [tex]\( x \rightarrow \infty \)[/tex]
1. Consider the term [tex]\( (1.25)^x \)[/tex]. When [tex]\( x \)[/tex] becomes a very large positive number, [tex]\( (1.25)^x \)[/tex] grows without bound.
2. This is because any base greater than 1 raised to a positive power results in a number that keeps increasing as the exponent increases.
So, as [tex]\( x \)[/tex] tends to [tex]\( \infty \)[/tex]:
[tex]\[ (1.25)^x \rightarrow \infty \][/tex]
Now substituting this into the function:
[tex]\[ w(x) = 5 \cdot (1.25)^x + 4 \][/tex]
[tex]\[ w(x) \rightarrow 5 \cdot \infty + 4 \][/tex]
[tex]\[ w(x) \rightarrow \infty \][/tex]
### Conclusion
Given our analysis, we can see that:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( w(x) \rightarrow 4 \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( w(x) \rightarrow \infty \)[/tex].
Thus, the correct statement describing the end behavior of the function is:
As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( w(x) \rightarrow 4 \)[/tex], and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( w(x) \rightarrow \infty \)[/tex].
