Let's analyze this problem and find the correct answer.

Problem Restatement

We are given a function $f$ where:

• For every increase in $x$ by $c$, where $c$ is a positive constant, $f(x)$ increases by a factor of 27.
• We want to determine which form of the function displays $\frac{1}{c}$ as a coefficient of $x$.

Step-by-Step Solution

This is a problem about exponential growth with a base raised to a power proportional to $x$. The key here is recognizing how the factor of growth relates to $x$ when it increases by increments of $c$.

Step 1: Identify the factor of increase

• We are told that $f(x)$ increases by a factor of 27 for every increase in $x$ by $c$. This implies: $f(x + c) = 27 \cdot f(x)$ Since this is an exponential function, we expect a base to be raised to a power involving $x$, such as: $f(x) = a \cdot b^{x/c},$ where $b$ represents the base (growth factor), and $a$ is a constant. We need to match this form to one of the provided choices.

Step 2: Express the base and the factor of 27

$b^{c/c} = b = 27 \quad \Rightarrow \quad b = 3^3.$

Thus, the equivalent form will have $3^{x/c}$ as part of the function.

Step 3: Check each answer option

We want the form that displays $\frac{1}{c}$ in the exponent for $x$:

• Option A: $f(x) = 48(3)^{\frac{1}{2}x}$. This shows a base of $3$, but it is scaled with $x$ by $\frac{1}{2}$, which does not match our problem’s increase by a factor of 27 for every $c$.
• Option B: $f(x) = 48(3^3)^{\frac{1}{c}x}$. This matches our derived form, where the factor $3^3 = 27$. So, this is the correct choice.
• Option C: $f(x) = 48(9)^{\frac{1}{4}x}$ shows a factor of 9, not 27.
• Option D: $f(x) = 48(27^{\frac{3}{2}x})^{1/2}$, which has incorrect scaling in the exponent.

Final Answer

The correct form is Option B:
$f(x) = 48(3^3)^{\frac{1}{c}x}.$

Would you like more details about exponential growth functions or specific steps in this explanation?
Here are 5 related questions:

1. How do you derive exponential functions from given growth factors?
2. Why is the base $3^3 = 27$ in this problem?
3. How does changing the value of $c$ affect the function’s growth rate?
4. What is the relationship between logarithmic and exponential growth problems?
5. How would the function change if the factor of growth were different, say, 81?

Tip

When dealing with exponential problems, look for key words like "factor of increase" to understand the base of the power function.