Answer :
To answer the question, let's start by finding the [tex]\( y \)[/tex]-intercepts of the given functions, [tex]\( g(x) = x + 2 \)[/tex] and [tex]\( f(x) = x - 1 \)[/tex].
### Step-by-Step Solution
1. Find the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
To find the [tex]\( y \)[/tex]-intercept, we need to set [tex]\( x = 0 \)[/tex] in the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 0 + 2 = 2 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is 2.
2. Find the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:
Similarly, we set [tex]\( x = 0 \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 0 - 1 = -1 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-1\)[/tex].
3. Calculate the difference in [tex]\( y \)[/tex]-intercepts:
To determine how many units below the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is, we subtract the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] from the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Difference in \( y \)-intercepts} = 2 - (-1) = 2 + 1 = 3 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is 3 units below the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex].
Based on this analysis, the correct answer is:
- 3 units
### Step-by-Step Solution
1. Find the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
To find the [tex]\( y \)[/tex]-intercept, we need to set [tex]\( x = 0 \)[/tex] in the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 0 + 2 = 2 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is 2.
2. Find the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:
Similarly, we set [tex]\( x = 0 \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 0 - 1 = -1 \][/tex]
So, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-1\)[/tex].
3. Calculate the difference in [tex]\( y \)[/tex]-intercepts:
To determine how many units below the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is, we subtract the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] from the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Difference in \( y \)-intercepts} = 2 - (-1) = 2 + 1 = 3 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is 3 units below the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex].
Based on this analysis, the correct answer is:
- 3 units
