To determine which of the given exponential functions is an increasing function, we need to analyze the nature of each function in terms of their bases. An exponential function [tex]\( f(x) = a^x \)[/tex] is increasing if the base [tex]\( a \)[/tex] is greater than 1.
Let's examine each of the functions:
1. [tex]\( f(x) = \left(\frac{3}{5}\right)^x \)[/tex]
- Here, the base is [tex]\( \frac{3}{5} \)[/tex], which is less than 1.
- [tex]\( \frac{3}{5} \approx 0.6 \)[/tex]
- Since [tex]\( 0.6 < 1 \)[/tex], this function is decreasing.
2. [tex]\( f(x) = 0.8^x \)[/tex]
- Here, the base is [tex]\( 0.8 \)[/tex], which is less than 1.
- Since [tex]\( 0.8 < 1 \)[/tex], this function is decreasing.
3. [tex]\( f(x) = 0.2^x \)[/tex]
- Here, the base is [tex]\( 0.2 \)[/tex], which is less than 1.
- Since [tex]\( 0.2 < 1 \)[/tex], this function is decreasing.
4. [tex]\( f(x) = \left(\frac{4}{3}\right)^x \)[/tex]
- Here, the base is [tex]\( \frac{4}{3} \)[/tex], which is greater than 1.
- [tex]\( \frac{4}{3} \approx 1.33 \)[/tex]
- Since [tex]\( 1.33 > 1 \)[/tex], this function is increasing.
Based on this analysis, the correct answer is:
[tex]\( f(x) = \left(\frac{4}{3}\right)^x \)[/tex]
This is the only function with a base greater than 1, making it an increasing exponential function.
