Let's carefully analyze the given functions and determine how they relate to each other.
Given the two functions:
[tex]\[ f(x) = e^x - 4 \][/tex]
[tex]\[ g(x) = \frac{1}{2} e^x - 4 \][/tex]
We'll compare the effect of the coefficient [tex]\( \frac{1}{2} \)[/tex] in [tex]\( g(x) \)[/tex] relative to [tex]\( f(x) \)[/tex].
### Step-by-step analysis:
1. Identify the exponential component: Both functions involve the exponential function [tex]\( e^x \)[/tex].
2. Vertical transformation:
- In [tex]\( f(x) \)[/tex], the exponential function [tex]\( e^x \)[/tex] is directly used.
- In [tex]\( g(x) \)[/tex], the exponential function [tex]\( e^x \)[/tex] is multiplied by [tex]\( \frac{1}{2} \)[/tex].
Let's note what happens when the exponential function is multiplied by a constant less than 1. Multiplying by [tex]\( \frac{1}{2} \)[/tex] compresses the graph vertically because each output value of [tex]\( e^x \)[/tex] is halved before applying the vertical shift.
3. Vertical shift (Subtraction of 4):
- Both functions have [tex]\( -4 \)[/tex] subtracted from the result, which means both graphs are shifted downward by 4 units.
- This shift does not affect the comparison we are making regarding stretching or compression between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
### Conclusion:
- The key difference between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is the multiplication of [tex]\( e^x \)[/tex] by [tex]\( \frac{1}{2} \)[/tex]. This multiplication compresses the graph vertically because all y-values are reduced by half.
Hence, the accurate description of how [tex]\( g(x) \)[/tex] relates to [tex]\( f(x) \)[/tex] is that the graph of [tex]\( g(x) \)[/tex] is a vertical compression of the graph of [tex]\( f(x) \)[/tex].
Therefore, the correct statement is:
[tex]\[ \boxed{\text{C. The graph of function } g \text{ is a vertical compression of the graph of function } f.} \][/tex]
