To understand the transformation of the function from [tex]\( f(x) = 3^x \)[/tex] to [tex]\( g(x) = \frac{1}{2} \cdot 3^x \)[/tex], let's analyze the effect of this transformation step-by-step.
1. Understanding the Original Function [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = 3^x \)[/tex] is an exponential function, where the base [tex]\( 3 \)[/tex] is raised to the power of [tex]\( x \)[/tex].
2. Understanding the Transformed Function [tex]\( g(x) \)[/tex]:
- The transformed function is given by [tex]\( g(x) = \frac{1}{2} \cdot 3^x \)[/tex].
- Here, [tex]\( \frac{1}{2} \)[/tex] is multiplied to the entire function [tex]\( 3^x \)[/tex].
3. Analyzing the Effect of Multiplying by [tex]\( \frac{1}{2} \)[/tex]:
- When we multiply a function by a constant factor [tex]\( a \)[/tex], the effect depends on the value of [tex]\( a \)[/tex].
- If [tex]\( 0 < a < 1 \)[/tex] (which is the case here since [tex]\( a = \frac{1}{2} \)[/tex]), the function is compressed vertically. This means the graph of the function is "squashed" towards the x-axis by the factor [tex]\( a \)[/tex].
- Conversely, if [tex]\( a > 1 \)[/tex], the function is stretched vertically.
4. Applying This to Our Function:
- For the function [tex]\( g(x) = \frac{1}{2} \cdot 3^x \)[/tex], the constant factor [tex]\( a = \frac{1}{2} \)[/tex] compresses the graph of [tex]\( 3^x \)[/tex] vertically by a factor of [tex]\( \frac{1}{2} \)[/tex].
5. Conclusion:
- Therefore, the effect of dilating the function [tex]\( f(x) = 3^x \)[/tex] to become [tex]\( g(x) = \frac{1}{2} \cdot 3^x \)[/tex] is a vertical compression by a factor of [tex]\( \frac{1}{2} \)[/tex].
Hence, the correct choice among the provided options is:
[tex]\[ f(x) \text{ is compressed vertically by a factor of } \frac{1}{2}. \][/tex]
So, the answer is [tex]\( \boxed{4} \)[/tex].
