Let's analyze the given function step-by-step:
The function provided is [tex]\( r(x) = 3 \cdot (0.75)^x - 8 \)[/tex].
### End Behavior as [tex]\( x \to -\infty \)[/tex]
1. Base Analysis: The base of the exponential part is [tex]\( 0.75 \)[/tex]. Note that [tex]\( 0.75 \)[/tex] is a fraction less than 1.
2. Behavior of [tex]\( (0.75)^x \)[/tex] as [tex]\( x \to -\infty \)[/tex]: When [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex], [tex]\( (0.75)^x \)[/tex] behaves as follows:
- For large negative [tex]\( x \)[/tex], the exponent [tex]\( x \)[/tex] makes [tex]\( 0.75^x \)[/tex] effectively an increasingly large number, because any fraction [tex]\( 0 < a < 1 \)[/tex] raised to a very large negative power becomes a very large positive number.
3. Result of [tex]\( r(x) \)[/tex] as [tex]\( x \to -\infty \)[/tex]: Hence, [tex]\( 3 \cdot (0.75)^x \)[/tex] will trend towards infinity, and subtracting 8 has negligible effect on it, so [tex]\( r(x) \)[/tex] approaches infinity.
Conclusively, as [tex]\( x \to -\infty \)[/tex], [tex]\( r(x) \to \infty \)[/tex].
### End Behavior as [tex]\( x \to \infty \)[/tex]
1. Behavior of [tex]\( (0.75)^x \)[/tex] as [tex]\( x \to \infty \)[/tex]: When [tex]\( x \)[/tex] approaches [tex]\(\infty\)[/tex], [tex]\( (0.75)^x \)[/tex] turns into a very small positive number because:
- Raising a fraction less than 1 to a large power results in a number approaching 0.
2. Result of [tex]\( r(x) \)[/tex] as [tex]\( x \to \infty \)[/tex]: Hence, [tex]\( 3 \cdot (0.75)^x \)[/tex] turns nearly 0. Subtracting 8 gives:
[tex]\[ r(x) \approx 0 - 8 = -8 \][/tex]
Conclusively, as [tex]\( x \to \infty \)[/tex], [tex]\( r(x) \to -8 \)[/tex].
### Correct Statement
From the analysis, the correct end behavior description is:
As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( r(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( r(x) \rightarrow -8 \)[/tex].
Hence, the correct answer is:
- As [tex]\( x \rightarrow -\infty, r(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow \infty, r(x) \rightarrow -8 \)[/tex].
