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The balances in two separate bank accounts that grow each month at different rates are represented by the functions [tex]f(x)[/tex] and [tex]g(x)[/tex]. In what month do the funds in the [tex]f(x)[/tex] bank account exceed those in the [tex]g(x)[/tex] bank account?

\begin{tabular}{|l|l|l|}
\hline
Month [tex](x)[/tex] & [tex]f(x) = 2^x[/tex] & [tex]g(x) = 4x + 12[/tex] \\
\hline
1 & 2 & 16 \\
\hline
2 & 4 & 20 \\
\hline
\end{tabular}

A. Month 3

B. Month 4

C. Month 5

D. Month 6

Sagot :

GinnyAnswer
To determine when the funds in the bank account represented by [tex]\( f(x) \)[/tex] exceed those in the bank account represented by [tex]\( g(x) \)[/tex], we can follow these steps:

1. Understand the functions:
- [tex]\( f(x) = 2^x \)[/tex] represents the balance in the first bank account.
- [tex]\( g(x) = 4x + 12 \)[/tex] represents the balance in the second bank account.

2. Compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] for different months:
- We need to find the smallest integer [tex]\( x \)[/tex] where [tex]\( f(x) > g(x) \)[/tex].

3. Evaluate both functions at different integer values of [tex]\(x\)[/tex]:
- Month 1 ([tex]\( x = 1 \)[/tex]):
- [tex]\( f(1) = 2^1 = 2 \)[/tex]
- [tex]\( g(1) = 4 \times 1 + 12 = 16 \)[/tex]
- [tex]\( f(1) = 2 \)[/tex], [tex]\( g(1) = 16 \)[/tex], so [tex]\( f(1) < g(1) \)[/tex].

- Month 2 ([tex]\( x = 2 \)[/tex]):
- [tex]\( f(2) = 2^2 = 4 \)[/tex]
- [tex]\( g(2) = 4 \times 2 + 12 = 20 \)[/tex]
- [tex]\( f(2) = 4 \)[/tex], [tex]\( g(2) = 20 \)[/tex], so [tex]\( f(2) < g(2) \)[/tex].

- Month 3 ([tex]\( x = 3 \)[/tex]):
- [tex]\( f(3) = 2^3 = 8 \)[/tex]
- [tex]\( g(3) = 4 \times 3 + 12 = 24 \)[/tex]
- [tex]\( f(3) = 8 \)[/tex], [tex]\( g(3) = 24 \)[/tex], so [tex]\( f(3) < g(3) \)[/tex].

- Month 4 ([tex]\( x = 4 \)[/tex]):
- [tex]\( f(4) = 2^4 = 16 \)[/tex]
- [tex]\( g(4) = 4 \times 4 + 12 = 28 \)[/tex]
- [tex]\( f(4) = 16 \)[/tex], [tex]\( g(4) = 28 \)[/tex], so [tex]\( f(4) < g(4) \)[/tex].

- Month 5 ([tex]\( x = 5 \)[/tex]):
- [tex]\( f(5) = 2^5 = 32 \)[/tex]
- [tex]\( g(5) = 4 \times 5 + 12 = 32 \)[/tex]
- [tex]\( f(5) = 32 \)[/tex], [tex]\( g(5) = 32 \)[/tex], so [tex]\( f(5) = g(5) \)[/tex].

- Month 6 ([tex]\( x = 6 \)[/tex]):
- [tex]\( f(6) = 2^6 = 64 \)[/tex]
- [tex]\( g(6) = 4 \times 6 + 12 = 36 \)[/tex]
- [tex]\( f(6) = 64 \)[/tex], [tex]\( g(6) = 36 \)[/tex], so [tex]\( f(6) > g(6) \)[/tex].

Therefore, the funds in the bank account represented by [tex]\( f(x) \)[/tex] exceed those in the bank account represented by [tex]\( g(x) \)[/tex] in month 6.
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