To determine which of the given functions represent exponential decay, we need to recall that exponential decay occurs when the function's base is raised to a negative exponent or if the base is a fraction between 0 and 1. Let's analyze each function one by one:
1. [tex]\( f(x) = 3(1.7)^{x-2} \)[/tex]
In this function, the base is [tex]\( 1.7 \)[/tex], which is greater than 1. The exponent is [tex]\( x-2 \)[/tex], and since the base is greater than 1, this function does not represent exponential decay because there is no negative exponent to cause the value of the function to decrease.
2. [tex]\( f(x) = 3(1.7)^{-2x} \)[/tex]
In this function, the base is [tex]\( 1.7 \)[/tex], which is greater than 1, but the exponent is [tex]\( -2x \)[/tex]. The negative exponent causes the function to decay because any positive value raised to a negative exponent results in a decreasing function.
3. [tex]\( f(x) = 3^5\left(\frac{1}{3}\right)^x \)[/tex]
In this function, the base is [tex]\( \frac{1}{3} \)[/tex], which is a fraction between 0 and 1. Therefore, as [tex]\( x \)[/tex] increases, [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] decreases, indicating that this function represents exponential decay.
4. [tex]\( f(x) = 3^5(2)^{-x} \)[/tex]
In this function, the base is [tex]\( 2 \)[/tex], which is greater than 1, but the exponent is [tex]\( -x \)[/tex]. The negative exponent causes the function to decay because any positive value raised to a negative exponent results in a decreasing function.
Summarizing the analysis:
- [tex]\( f(x)=3(1.7)^{x-2} \)[/tex] does not represent exponential decay.
- [tex]\( f(x)=3(1.7)^{-2x} \)[/tex] represents exponential decay.
- [tex]\( f(x)=3^5\left(\frac{1}{3}\right)^x \)[/tex] represents exponential decay.
- [tex]\( f(x)=3^5(2)^{-x} \)[/tex] represents exponential decay.
Therefore, the functions that represent exponential decay are:
1. [tex]\( f(x)=3(1.7)^{-2x} \)[/tex]
2. [tex]\( f(x)=3^5\left(\frac{1}{3}\right)^x \)[/tex]
3. [tex]\( f(x)=3^5(2)^{-x} \)[/tex]
