Let's analyze the end behavior of the function [tex]\( f(x) = 10(0.75)^x \)[/tex].
1. Left End Behavior (as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]):
- When [tex]\( x \)[/tex] becomes a large negative number (i.e., [tex]\( x \rightarrow -\infty \)[/tex]), the term [tex]\((0.75)^x\)[/tex] behaves differently.
- Raising 0.75 to a large negative power results in a very large positive number because 0.75 is a fraction less than 1, and raising it to negative powers flips it into large positive numbers.
- Therefore, multiplying this large positive number by 10 results in the function [tex]\( f(x) \)[/tex] becoming very large.
Hence, as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches positive infinity.
2. Right End Behavior (as [tex]\( x \)[/tex] approaches [tex]\( \infty\)[/tex]):
- When [tex]\( x \)[/tex] becomes a large positive number (i.e., [tex]\( x \rightarrow \infty \)[/tex]), the term [tex]\((0.75)^x\)[/tex] decreases towards 0 because raising a fraction (less than 1) to a large positive power makes it smaller and smaller.
- Therefore, multiplying this small number by 10 results in the function [tex]\( f(x) \)[/tex] approaching 0.
Hence, as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
Combining both parts, we can describe the end behavior of the function [tex]\( f(x) = 10(0.75)^x \)[/tex] as follows:
The left end approaches [tex]\(\boxed{\text{positive infinity}}\)[/tex], and the right end approaches [tex]\(\boxed{0}\)[/tex].
