To determine which functions decrease at a rate of 40% per unit increase in [tex]\( x \)[/tex], we need to analyze the given functions and their behavior as [tex]\( x \)[/tex] increases.
Let's examine each function to understand their respective rates of decrease:
1. Function A: [tex]\( f(x) = \frac{3}{2}\left(\frac{2}{5}\right)^x \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = \frac{3}{2} \left(\frac{2}{5}\right)^0 = \frac{3}{2} \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = \frac{3}{2} \left(\frac{2}{5}\right) = \frac{3}{2} \cdot \frac{2}{5} = \frac{6}{10} = 0.6 \)[/tex]
- The ratio [tex]\( \frac{f(1)}{f(0)} = \frac{0.6}{1.5} = 0.4 \)[/tex]
2. Function B: [tex]\( f(x) = \frac{2}{3}\left(\frac{3}{5}\right)^x \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = \frac{2}{3}\left(\frac{3}{5}\right)^0 = \frac{2}{3} \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = \frac{2}{3} \left(\frac{3}{5}\right) = \frac{2}{3} \cdot \frac{3}{5} = \frac{6}{15} = 0.4 \)[/tex]
3. Function C: [tex]\( f(x) = \frac{2}{5}\left(\frac{5}{8}\right)^x \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = \frac{2}{5}\left(\frac{5}{8}\right)^0 = \frac{2}{5} \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = \frac{2}{5} \left(\frac{5}{8}\right) = \frac{2}{5} \cdot \frac{5}{8} = \frac{10}{40} = 0.25 \)[/tex]
4. Function D: [tex]\( f(x) = 1.75(0.6)^x \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 1.75(0.6)^0 = 1.75 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 1.75(0.6) = 1.05 \)[/tex]
- The ratio [tex]\( \frac{f(1)}{f(0)} = \frac{1.05}{1.75} = 0.6 \)[/tex]
5. Function E: [tex]\( f(x) = 2.5(0.4)^x \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 2.5(0.4)^0 = 2.5 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 2.5(0.4) = 1 \)[/tex]
- The ratio [tex]\( \frac{f(1)}{f(0)} = \frac{1}{2.5} = 0.4 \)[/tex]
6. Function F: [tex]\( f(x) = 0.4(0.8)^x \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 0.4(0.8)^0 = 0.4 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 0.4(0.8) = 0.32 \)[/tex]
- The ratio [tex]\( \frac{f(1)}{f(0)} = \frac{0.32}{0.4} = 0.8 \)[/tex]
After analyzing all the given functions, the ones that decrease at a rate of 40% per unit increase in [tex]\( x \)[/tex] are:
- Function A: [tex]\( f(x) = \frac{3}{2}\left(\frac{2}{5}\right)^x \)[/tex]
- Function E: [tex]\( f(x) = 2.5(0.4)^x \)[/tex]
Thus, the functions that match the criteria are:
[tex]\[ \text{A and E} \][/tex]
