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Sagot :
To determine which function represents [tex]\( g(x) \)[/tex] given that it has a [tex]\( y \)[/tex]-intercept of [tex]\(-2\)[/tex], let’s analyze each of the provided functions step-by-step.
### Step 1: Understanding the Y-Intercept
The [tex]\( y \)[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex]. Therefore, we need to check what [tex]\( g(0) \)[/tex] is for each given option.
### Step 2: Analyzing Each Transformation
#### Option A: [tex]\( g(x) = f(x+2) \)[/tex]
For [tex]\( g(x) = f(x+2) \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0+2) = f(2) \][/tex]
This transformation shifts the function horizontally but does not affect the [tex]\( y \)[/tex]-intercept of the original function [tex]\( f(x) \)[/tex] directly. Hence, the [tex]\( y \)[/tex]-intercept remains whatever it was for [tex]\( f(x) \)[/tex], not necessarily [tex]\(-2\)[/tex].
#### Option B: [tex]\( g(x) = f(x) - 5 \)[/tex]
For [tex]\( g(x) = f(x) - 5 \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0) - 5 \][/tex]
This transformation shifts the entire function vertically down by 5 units. If the original [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( y_0 \)[/tex], then the new [tex]\( y \)[/tex]-intercept will be [tex]\( y_0 - 5 \)[/tex]. This does not guarantee a [tex]\( y \)[/tex]-intercept of [tex]\(-2\)[/tex] unless [tex]\( y_0 \)[/tex] was [tex]\(3\)[/tex].
#### Option C: [tex]\( g(x) = f(x-5) \)[/tex]
For [tex]\( g(x) = f(x-5) \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0-5) = f(-5) \][/tex]
This transformation shifts the function horizontally but does not affect the [tex]\( y \)[/tex]-intercept of the original function [tex]\( f(x) \)[/tex] directly. Hence, the [tex]\( y \)[/tex]-intercept remains whatever it was for [tex]\( f(x) \)[/tex], not necessarily [tex]\(-2\)[/tex].
#### Option D: [tex]\( g(x) = f(x) - 2 \)[/tex]
For [tex]\( g(x) = f(x) - 2 \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0) - 2 \][/tex]
This transformation shifts the entire function vertically down by 2 units. If the original [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( y_0 \)[/tex], then the new [tex]\( y \)[/tex]-intercept will be [tex]\( y_0 - 2 \)[/tex]. Therefore, if the original [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] was [tex]\( 0 \)[/tex], the new [tex]\( y \)[/tex]-intercept becomes [tex]\( -2 \)[/tex], which matches the given condition.
### Conclusion
The only transformation that results in a [tex]\( y \)[/tex]-intercept of [tex]\(-2\)[/tex] is option D:
[tex]\[ \boxed{g(x) = f(x) - 2} \][/tex]
### Step 1: Understanding the Y-Intercept
The [tex]\( y \)[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex]. Therefore, we need to check what [tex]\( g(0) \)[/tex] is for each given option.
### Step 2: Analyzing Each Transformation
#### Option A: [tex]\( g(x) = f(x+2) \)[/tex]
For [tex]\( g(x) = f(x+2) \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0+2) = f(2) \][/tex]
This transformation shifts the function horizontally but does not affect the [tex]\( y \)[/tex]-intercept of the original function [tex]\( f(x) \)[/tex] directly. Hence, the [tex]\( y \)[/tex]-intercept remains whatever it was for [tex]\( f(x) \)[/tex], not necessarily [tex]\(-2\)[/tex].
#### Option B: [tex]\( g(x) = f(x) - 5 \)[/tex]
For [tex]\( g(x) = f(x) - 5 \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0) - 5 \][/tex]
This transformation shifts the entire function vertically down by 5 units. If the original [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( y_0 \)[/tex], then the new [tex]\( y \)[/tex]-intercept will be [tex]\( y_0 - 5 \)[/tex]. This does not guarantee a [tex]\( y \)[/tex]-intercept of [tex]\(-2\)[/tex] unless [tex]\( y_0 \)[/tex] was [tex]\(3\)[/tex].
#### Option C: [tex]\( g(x) = f(x-5) \)[/tex]
For [tex]\( g(x) = f(x-5) \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0-5) = f(-5) \][/tex]
This transformation shifts the function horizontally but does not affect the [tex]\( y \)[/tex]-intercept of the original function [tex]\( f(x) \)[/tex] directly. Hence, the [tex]\( y \)[/tex]-intercept remains whatever it was for [tex]\( f(x) \)[/tex], not necessarily [tex]\(-2\)[/tex].
#### Option D: [tex]\( g(x) = f(x) - 2 \)[/tex]
For [tex]\( g(x) = f(x) - 2 \)[/tex], when we substitute [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = f(0) - 2 \][/tex]
This transformation shifts the entire function vertically down by 2 units. If the original [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( y_0 \)[/tex], then the new [tex]\( y \)[/tex]-intercept will be [tex]\( y_0 - 2 \)[/tex]. Therefore, if the original [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] was [tex]\( 0 \)[/tex], the new [tex]\( y \)[/tex]-intercept becomes [tex]\( -2 \)[/tex], which matches the given condition.
### Conclusion
The only transformation that results in a [tex]\( y \)[/tex]-intercept of [tex]\(-2\)[/tex] is option D:
[tex]\[ \boxed{g(x) = f(x) - 2} \][/tex]
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