The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are described using the following equation and table:

[tex]\[ f(x) = -6(1.02)^x \][/tex]

[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-1 & -5 \\
\hline
0 & -3 \\
\hline
1 & -1 \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]

Which equation best compares the [tex]\( y \)[/tex]-intercepts of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]?

A. The [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is equal to the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex].

B. 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].

C. 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].

D. 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].



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]