To determine the end behavior of the given function [tex]\( f(x) = 10 \cdot (0.75)^x \)[/tex], we need to consider what happens to the values of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches negative infinity and positive infinity.
### Left End Behavior
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
- The base of the exponential function, [tex]\( 0.75 \)[/tex], is a fraction less than 1.
- When raised to a large negative power, a fraction less than 1 becomes very large.
- Hence, [tex]\( (0.75)^x \)[/tex] becomes very large as [tex]\( x \)[/tex] becomes more negative.
- Multiplying by 10 does not change the fundamental behavior; thus, [tex]\( f(x) \)[/tex] approaches positive infinity.
### Right End Behavior
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]):
- The base [tex]\( 0.75 \)[/tex] raised to a large positive power gets smaller and smaller, approaching zero.
- Therefore, [tex]\( (0.75)^x \)[/tex] approaches zero as [tex]\( x \)[/tex] increases.
- Multiplying by 10 still results in the function approaching zero.
To summarize:
- The left end approaches positive infinity.
- The right end approaches zero.
Therefore, the completed statement is:
"The left end approaches [tex]\( \infty \)[/tex] and the right end approaches [tex]\( 0 \)[/tex]."
