Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
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.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.