To compare the [tex]$y$[/tex]-intercepts of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we follow these steps:
1. Find the [tex]$y$[/tex]-intercept of [tex]\(f(x)\)[/tex]:
To determine the [tex]$y$[/tex]-intercept of [tex]\(f(x)\)[/tex], we evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex].
[tex]\[
f(0) = -6(1.05)^0
\][/tex]
Since any number raised to the power of 0 is 1, this simplifies to:
[tex]\[
f(0) = -6 \cdot 1 = -6.0
\][/tex]
Therefore, the [tex]$y$[/tex]-intercept of [tex]\(f(x)\)[/tex] is [tex]\(-6.0\)[/tex].
2. Find the [tex]$y$[/tex]-intercept of [tex]\(g(x)\)[/tex]:
To determine the [tex]$y$[/tex]-intercept of [tex]\(g(x)\)[/tex], we look at the value of [tex]\(g(x)\)[/tex] corresponding to [tex]\(x = 0\)[/tex] in the provided table.
[tex]\[
g(0) = -3
\][/tex]
Therefore, the [tex]$y$[/tex]-intercept of [tex]\(g(x)\)[/tex] is [tex]\(-3\)[/tex].
3. Compare the [tex]$y$[/tex]-intercepts:
The [tex]$y$[/tex]-intercept of [tex]\(f(x)\)[/tex] is [tex]\(-6.0\)[/tex], and the [tex]$y$[/tex]-intercept of [tex]\(g(x)\)[/tex] is [tex]\(-3\)[/tex]. To find the best comparison:
[tex]\[
\text{The $y$-intercept of \(g(x)\), which is \(-3\), is equal to 2 times the $y$-intercept of \(f(x)\), which is \(-6.0\)}
\][/tex]
[tex]\[
-3 = 2 \times (-6.0)
\][/tex]
Therefore, the correct comparison according to the given [tex]\(y\)[/tex]-intercepts is:
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].
