Answer :
To determine which equation best compares the [tex]$y$[/tex]-intercepts of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we need to find the [tex]$y$[/tex]-intercepts of both functions and then compare them.
1. Determine the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex]:
The [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is found by evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -6 \cdot (1.02)^0 \][/tex]
Since [tex]\( (1.02)^0 = 1 \)[/tex]:
[tex]\[ f(0) = -6 \cdot 1 = -6 \][/tex]
Thus, the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( -6 \)[/tex].
2. Determine the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex]:
From the given table, the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex] is [tex]\( -3 \)[/tex].
[tex]\[ g(0) = -3 \][/tex]
Thus, the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\( -3 \)[/tex].
3. Compare the [tex]$y$[/tex]-intercepts:
- [tex]\( y \)[/tex]-intercept of [tex]\( f(x) = -6 \)[/tex]
- [tex]\( y \)[/tex]-intercept of [tex]\( g(x) = -3 \)[/tex]
Now, we need to check the given comparisons:
- The [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is equal to the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex].
[tex]\[ -6 = -3 \quad \text{(False)} \][/tex]
- The [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is equal to 2 times the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex].
[tex]\[ -6 = 2 \cdot (-3) = -6 \quad \text{(True)} \][/tex]
- The [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex] is equal to 2 times the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex].
[tex]\[ -3 = 2 \cdot (-6) = -12 \quad \text{(False)} \][/tex]
- The [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex] is equal to 2 plus the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex].
[tex]\[ -3 = -6 + 2 = -4 \quad \text{(False)} \][/tex]
Therefore, the correct comparison is:
[tex]\[ \text{The $y$-intercept of \( f(x) \) is equal to 2 times the $y$-intercept of \( g(x) \)}. \][/tex]
1. Determine the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex]:
The [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is found by evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -6 \cdot (1.02)^0 \][/tex]
Since [tex]\( (1.02)^0 = 1 \)[/tex]:
[tex]\[ f(0) = -6 \cdot 1 = -6 \][/tex]
Thus, the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( -6 \)[/tex].
2. Determine the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex]:
From the given table, the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 0 \)[/tex] is [tex]\( -3 \)[/tex].
[tex]\[ g(0) = -3 \][/tex]
Thus, the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\( -3 \)[/tex].
3. Compare the [tex]$y$[/tex]-intercepts:
- [tex]\( y \)[/tex]-intercept of [tex]\( f(x) = -6 \)[/tex]
- [tex]\( y \)[/tex]-intercept of [tex]\( g(x) = -3 \)[/tex]
Now, we need to check the given comparisons:
- The [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is equal to the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex].
[tex]\[ -6 = -3 \quad \text{(False)} \][/tex]
- The [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex] is equal to 2 times the [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex].
[tex]\[ -6 = 2 \cdot (-3) = -6 \quad \text{(True)} \][/tex]
- The [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex] is equal to 2 times the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex].
[tex]\[ -3 = 2 \cdot (-6) = -12 \quad \text{(False)} \][/tex]
- The [tex]$y$[/tex]-intercept of [tex]\( g(x) \)[/tex] is equal to 2 plus the [tex]$y$[/tex]-intercept of [tex]\( f(x) \)[/tex].
[tex]\[ -3 = -6 + 2 = -4 \quad \text{(False)} \][/tex]
Therefore, the correct comparison is:
[tex]\[ \text{The $y$-intercept of \( f(x) \) is equal to 2 times the $y$-intercept of \( g(x) \)}. \][/tex]
