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) =

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