Answer:
1. h(x) = [tex]5(1-\frac{8}{9})^{x}[/tex]
2. g(x) = [tex]3(1-\frac{3}{4})^{x}[/tex]
3. f(x) = [tex](1-0.65)^{x}[/tex]
Step-by-step explanation:
The base of the exponential function is b, where
- r is the percentage of increase or decrease in decimal
- The exponential function is f(x) = [tex]a(b)^{x}[/tex] OR f(x) = [tex]a(1-+r)^{x}[/tex]
∵ h(x) = [tex]5(\frac{1}{9})^{x}[/tex]
→ Compare it by the form above
∴ b = [tex]\frac{1}{9}[/tex]
∵ b = 1 - r
∴ [tex]\frac{1}{9}[/tex] = 1 - r
→ Add r to both sides
∴ [tex]\frac{1}{9}[/tex] + r = 1 - r + r
∴ [tex]\frac{1}{9}[/tex] + r = 1
Subtract [tex]\frac{1}{9}[/tex] from both sides
∴ [tex]\frac{1}{9}[/tex] - [tex]\frac{1}{9}[/tex] + r = 1 - [tex]\frac{1}{9}[/tex]
∴ r = [tex]\frac{8}{9}[/tex]
∴ h(x) = [tex]5(1-\frac{8}{9})^{x}[/tex]
∵ g(x) = [tex]3(\frac{1}{4})^{x}[/tex]
→ Compare it by the form above
∴ b = [tex]\frac{1}{4}[/tex]
∵ b = 1 - r
∴ [tex]\frac{1}{4}[/tex] = 1 - r
→ Add r to both sides
∴ [tex]\frac{1}{4}[/tex] + r = 1 - r + r
∴ [tex]\frac{1}{4}[/tex] + r = 1
Subtract [tex]\frac{1}{4}[/tex] from both sides
∴ [tex]\frac{1}{4}[/tex] - [tex]\frac{1}{4}[/tex] + r = 1 - [tex]\frac{1}{4}[/tex]
∴ r = [tex]\frac{3}{4}[/tex]
∴ g(x) = [tex]3(1-\frac{3}{4})^{x}[/tex]
∵ f(x) = [tex](0.35)^{x}[/tex]
→ Compare it by the form above
∴ b = 0.35
∵ b = 1 - r
∴ 0.35 = 1 - r
→ Add r to both sides
∴ 0.35 + r = 1 - r + r
∴ 0.35 + r = 1
Subtract 0.35 from both sides
∴ 0.35 - 0.35 + r = 1 - 0.35
∴ r = 0.65
∴ f(x) = [tex](1-0.65)^{x}[/tex]
