Alright, let's determine which function represents a vertical stretch of an exponential function.
First, let's recall what a vertical stretch is. A vertical stretch occurs when all the [tex]\( y \)[/tex]-values of a function are multiplied by a constant factor greater than 1. This operation will make the graph of the function taller.
Now, let’s analyze each function provided:
1. [tex]\( f(x) = 3 \left( \frac{1}{2} \right)^x \)[/tex]
This function has a multiplier of 3 in front of the exponential part [tex]\( \left( \frac{1}{2} \right)^x \)[/tex]. This indicates a vertical stretch since multiplying the exponential part by 3 increases its height. So this function does represent a vertical stretch.
2. [tex]\( f(x) = \frac{1}{2} (3)^x \)[/tex]
Here, the function has a multiplier of [tex]\( \frac{1}{2} \)[/tex]. Multiplying by a factor between 0 and 1 does not stretch the function vertically. Rather, it compresses it because it reduces the height of the graph. So this function does not represent a vertical stretch.
3. [tex]\( \pi(x) = (3)^{2x} \)[/tex]
In this function, the base of the exponential is 3 and has an exponent of [tex]\( 2x \)[/tex]. This represents a change in the rate of growth of the exponential function but does not involve a vertical stretch which requires a constant multiplier outside the exponential term. So this function does not represent a vertical stretch.
4. [tex]\( f(x) = 3^{\left(\frac{1}{2} x\right)} \)[/tex]
This function involves an exponent that is [tex]\( \frac{1}{2} x \)[/tex]. Again, this changes the rate of growth or decay of the exponential function but still does not represent a vertical stretch via constant multiplication of the entire function. So this function does not represent a vertical stretch.
After reviewing all the options, we can conclude that option 1 provides a vertical stretch of the exponential function.
Thus, the correct function representing a vertical stretch is:
[tex]\[ f(x) = 3 \left( \frac{1}{2} \right)^x \][/tex]
Therefore, the answer is:
[tex]\[ 2 \][/tex]
