Answer :
To determine which statement is true, let's analyze the given functions step-by-step.
The functions are:
[tex]\[ f(x) = \log_{15} x \][/tex]
[tex]\[ g(x) = \frac{1}{2} \log_{15}(x + 4) \][/tex]
### Step 1: Analyze the Vertical Transformation
The original function [tex]\( f(x) = \log_{15} x \)[/tex] is transformed to [tex]\( g(x) = \frac{1}{2} \log_{15}(x + 4) \)[/tex]. The coefficient [tex]\(\frac{1}{2}\)[/tex] in front of the logarithm function in [tex]\(g(x)\)[/tex] indicates a vertical transformation. Specifically, multiplying a function by a factor less than 1 results in a vertical shrink by that factor. Therefore, [tex]\(g(x)\)[/tex] is vertically shrunk by a factor of [tex]\(\frac{1}{2}\)[/tex].
### Step 2: Analyze the Horizontal Shift
The inside argument of the logarithm in [tex]\(g(x)\)[/tex] is changed from [tex]\(x\)[/tex] to [tex]\(x + 4\)[/tex]. A shift of [tex]\(x + k\)[/tex] inside a function indicates a horizontal shift. Specifically, adding a number [tex]\(k\)[/tex] inside the function argument shifts the function to the left by [tex]\(k\)[/tex] units. Therefore, [tex]\(g(x)\)[/tex] represents a horizontal shift to the left by 4 units.
### Conclusion:
Combining these two observations, we see that [tex]\(g(x)\)[/tex] is:
- Vertically shrunk by a factor of [tex]\(\frac{1}{2}\)[/tex]
- Shifted to the left by 4 units
The statement that correctly describes these transformations is:
[tex]\[ \text{g(x) is shrunk vertically by a factor of } \frac{1}{2} \text{ and shifted to the left 4 units compared to f(x).} \][/tex]
Thus, the correct statement is:
[tex]\[ \boxed{3} \][/tex]
The functions are:
[tex]\[ f(x) = \log_{15} x \][/tex]
[tex]\[ g(x) = \frac{1}{2} \log_{15}(x + 4) \][/tex]
### Step 1: Analyze the Vertical Transformation
The original function [tex]\( f(x) = \log_{15} x \)[/tex] is transformed to [tex]\( g(x) = \frac{1}{2} \log_{15}(x + 4) \)[/tex]. The coefficient [tex]\(\frac{1}{2}\)[/tex] in front of the logarithm function in [tex]\(g(x)\)[/tex] indicates a vertical transformation. Specifically, multiplying a function by a factor less than 1 results in a vertical shrink by that factor. Therefore, [tex]\(g(x)\)[/tex] is vertically shrunk by a factor of [tex]\(\frac{1}{2}\)[/tex].
### Step 2: Analyze the Horizontal Shift
The inside argument of the logarithm in [tex]\(g(x)\)[/tex] is changed from [tex]\(x\)[/tex] to [tex]\(x + 4\)[/tex]. A shift of [tex]\(x + k\)[/tex] inside a function indicates a horizontal shift. Specifically, adding a number [tex]\(k\)[/tex] inside the function argument shifts the function to the left by [tex]\(k\)[/tex] units. Therefore, [tex]\(g(x)\)[/tex] represents a horizontal shift to the left by 4 units.
### Conclusion:
Combining these two observations, we see that [tex]\(g(x)\)[/tex] is:
- Vertically shrunk by a factor of [tex]\(\frac{1}{2}\)[/tex]
- Shifted to the left by 4 units
The statement that correctly describes these transformations is:
[tex]\[ \text{g(x) is shrunk vertically by a factor of } \frac{1}{2} \text{ and shifted to the left 4 units compared to f(x).} \][/tex]
Thus, the correct statement is:
[tex]\[ \boxed{3} \][/tex]
