Question

13 mark for review
an exponential function f is defined by f(x) = k(3)^x, where k is a positive constant. if f(-1) = 6 and f(2) = 162, what is the value of k?
a 6
b 7
c 17
d 18

Explanation:

Step1: Use \( f(-1) = 6 \) to form an equation

Substitute \( x = -1 \) into \( f(x)=k(3)^{x} \), we get \( f(-1)=k(3)^{-1} \). Since \( f(-1) = 6 \), then \( k\times\frac{1}{3}=6 \).

Step2: Solve for \( k \)

Multiply both sides of the equation \( \frac{k}{3}=6 \) by 3, we have \( k = 6\times3 = 18 \). We can also verify with \( f(2) \): substitute \( x = 2 \) and \( k = 18 \) into \( f(x) \), \( f(2)=18\times(3)^{2}=18\times9 = 162 \), which matches the given condition.

Answer:

D. 18