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.
