Question

If a function f ( x ) has values f ( 4 ) = 6 and f ( 8 ) = 18, use what you have learned about function patterns to find f ( 16 ) = if f ( x ) is: a.) Linear function: f ( 16 ) = b.) Power function: f ( 16 ) = c.) Exponential function: f ( 16 ) = d.) Logarithmic function: f ( 16 ) =

Answer 1

Answer:

a) f(16) = 42

b) f(16) = 54

c) f(16) = 162

d) f(16) = 30

Step-by-step explanation:

a) y = mx + b ∧ m = (f(8) - f(4))/(8-4) ⇒ m = (18 - 6)/(8 - 4) = 3

b = y - mx = 6 - 3(4) = 6 - 12 = - 6

f(16) = 3(16) - 6 = 42

b) y = kxⁿ ∧ f(4) = 6 = k4ⁿ ∧ f(8) = 18 = k8ⁿ ⇒ 18/6 = (k8ⁿ)/(k4ⁿ) ⇒ 3 = 2ⁿ

n = ㏑(3) / ㏑(2) ⇒ k = y/xⁿ ⇒ k = 6/4ⁿ = 2/3

f(16) = 2/3 × 16ⁿ = 54

c) y = aeᵇˣ ∧ f(4) = 6 = aeᵇ⁴ ∧ f(8) = 18 = aeᵇ⁸ ⇒ 18/6 = (aeᵇ⁸)/(aeᵇ⁴) ⇒ 3 = e⁴ᵇ

b = ㏑(3/4) ∧ a = y / eᵇˣ ⇒ a = 6 / e⁴ᵇ = 2

f(16) = 2eᵇ¹⁶ = 162

d) y = a㏑(bx) ∧ f(4) = 6 = a㏑(b4) ∧ f(8) = 18 = a㏑(b8)

⇒ 18 - 6 = a㏑(b8) - a㏑(b4) ⇒ 12 = a㏑(8b/4b) ⇒ a = 12 / ㏑(2)

f(4) = 6 = a㏑(4b) ⇒ b = (√2)/4

f(16) = a㏑(b16) = 30