To understand the end behavior of the given function [tex]\( f(x) = \left(\frac{2}{3}\right)^x - 2 \)[/tex], we need to analyze how the function behaves as [tex]\( x \)[/tex] either increases towards infinity or decreases towards negative infinity.
1. Analyze the base of the exponential function:
We start with the term [tex]\( \left(\frac{2}{3}\right)^x \)[/tex]. The fraction [tex]\( \frac{2}{3} \)[/tex] is less than 1.
2. Behavior as [tex]\( x \)[/tex] increases:
- When [tex]\( x \)[/tex] increases (i.e., [tex]\( x \rightarrow \infty \)[/tex]), [tex]\( \left(\frac{2}{3}\right)^x \)[/tex] becomes a very small positive number, approaching 0.
- Therefore, [tex]\( f(x) = \left(\frac{2}{3}\right)^x - 2 \)[/tex] approaches [tex]\( 0 - 2 = -2 \)[/tex].
- Thus, the function [tex]\( f(x) \)[/tex] approaches [tex]\(-2\)[/tex] as [tex]\( x \)[/tex] increases to infinity.
3. Behavior as [tex]\( x \)[/tex] decreases:
- When [tex]\( x \)[/tex] decreases (i.e., [tex]\( x \rightarrow -\infty \)[/tex]), [tex]\( \left(\frac{2}{3}\right)^-x = \left(\frac{3}{2}\right)^x \)[/tex], where [tex]\( \left(\frac{3}{2}\right) > 1 \)[/tex].
- As [tex]\( x \)[/tex] goes further negative, [tex]\( \left(\frac{3}{2}\right)^x \)[/tex] grows exponentially larger.
- Therefore, [tex]\( f(x) = \left(\frac{2}{3}\right)^x - 2 \)[/tex] becomes a very large positive number minus 2, which approaches positive infinity.
Given these analyses, we conclude that:
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] approaches -2,
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] approaches infinity.
Based on these observations, the correct answer to describe the end behavior of the function [tex]\[ f(x)=\left(\frac{2}{3}\right)^x - 2 \][/tex] is:
D. As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] approaches -2.
