Question

Please help me out I have a math test tmr so I won't be sleeping on time. the 30 minutes I could take to answer these is toooooooooooo muchhhh. I'm not spoilt i swear but just please helllppp

Answer 1

1. h(x) =
`5(1-(8)/(9))^(x)`

2. g(x) =
`3(1-(3)/(4))^(x)`

3. f(x) =
`(1-0.65)^(x)`

Explanation:

The base of the exponential function is b, where

• b = 1 ± r, where

• r is the percentage of increase or decrease in decimal

• The exponential function is f(x) =
  `a(b)^(x)` OR f(x) =
  `a(1-+r)^(x)`

∵ h(x) =
`5((1)/(9))^(x)`

→ Compare it by the form above

∴ b =
`(1)/(9)`

∵ b = 1 - r

∴
`(1)/(9)` = 1 - r

→ Add r to both sides

∴
`(1)/(9)` + r = 1 - r + r

∴
`(1)/(9)` + r = 1

Subtract
`(1)/(9)` from both sides

∴
`(1)/(9)` -
`(1)/(9)` + r = 1 -
`(1)/(9)`

∴ r =
`(8)/(9)`

∴ h(x) =
`5(1-(8)/(9))^(x)`

∵ g(x) =
`3((1)/(4))^(x)`

→ Compare it by the form above

∴ b =
`(1)/(4)`

∵ b = 1 - r

∴
`(1)/(4)` = 1 - r

→ Add r to both sides

∴
`(1)/(4)` + r = 1 - r + r

∴
`(1)/(4)` + r = 1

Subtract
`(1)/(4)` from both sides

∴
`(1)/(4)` -
`(1)/(4)` + r = 1 -
`(1)/(4)`

∴ r =
`(3)/(4)`

∴ g(x) =
`3(1-(3)/(4))^(x)`

∵ f(x) =
`(0.35)^(x)`

→ 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) =
`(1-0.65)^(x)`