Question

Find the exponential function that contains the points (2,36) and (3,108).

Answer 1

Answer:

$f(x) = 4\times (3)^{x}$.

Step-by-step explanation:

Exponential functions are in the form $f(x) = a\, \left(b^{x}\right)$,  where $a$ and $b$ are constants with $b > 0$.

In this question, the function should satisfy that $f(2) = 36$ and $f(3) = 108$. Hence, constants $a$ and $b$ should ensure that:

$a \cdot b^2 = f(2) = 36$, and

$a \cdot b^3 = 108$.

Rewrite the left-hand side of the second equation: $a \cdot b^{3} = a \cdot b^2 \cdot b$.

Substitute the first equation $a \cdot b^{2} = 36$ into the second:

$a \cdot b^2 \cdot b = 108$.

$36\, b = 108$.

$\displaystyle b = \frac{108}{36} = 3$.

Substitute $b = 3$ into either equation (for example, the first equation) and solve for $a$:

$a \cdot 3^2 = 36$.

$a = 4$.

Hence, the exponential function $f(x) = a\cdot b^{x} = 4 \times (3)^{x}$ should satisfy the requirements of this question.