Simplify the function \( f(x)=\frac{1}{3}(81)^{\frac{3 x}{4}} \). Then determine the key aspects of the function. The initial value is The simplified base is The domain is The range is

Let's simplify the function and identify its key aspects step by step. ### Given Function \[ f(x) = \frac{1}{3} \cdot 81^{\frac{3x}{4}} \] ### 1. Simplifying the Function **Step 1:** Express 81 as a power of 3. \[ 81 = 3^4 \] \[ \therefore f(x) = \frac{1}{3} \cdot (3^4)^{\frac{3x}{4}} \] **Step 2:** Apply the power of a power rule \((a^m)^n = a^{m \cdot n}\). \[ f(x) = \frac{1}{3} \cdot 3^{4 \cdot \frac{3x}{4}} \] \[ f(x) = \frac{1}{3} \cdot 3^{3x} \] **Step 3:** Combine the coefficients with the same base using the rule \(a^m \cdot a^n = a^{m+n}\). \[ f(x) = 3^{-1} \cdot 3^{3x} \] \[ f(x) = 3^{3x - 1} \] **Simplified Function:** \[ f(x) = 3^{3x - 1} \] ### 2. Key Aspects of the Function - **Initial Value:** The initial value is the value of the function when \(x = 0\). \[ f(0) = 3^{3(0) - 1} = 3^{-1} = \frac{1}{3} \] - **Simplified Base:** The base of the simplified exponential function is **3**. - **Domain:** Exponential functions are defined for all real numbers. \[ \text{Domain: } (-\infty, \infty) \] - **Range:** Exponential functions with positive bases greater than 1 have outputs that are always positive. \[ \text{Range: } (0, \infty) \] ### Summary - **Simplified Function:** \[ f(x) = 3^{3x - 1} \] - **Initial Value:** \[ f(0) = \frac{1}{3} \] - **Simplified Base:** \[ 3 \] - **Domain:** \[ (-\infty, \infty) \] - **Range:** \[ (0, \infty) \]