Simplify the function \( f(x)=\frac{1}{2}(27)^{\frac{2 x}{3}} \). Then determine the key aspects of the function. The initial value is The simplified base is The domain is The range is

To simplify the function \( f(x) = \frac{1}{2} \cdot 27^{\frac{2x}{3}} \) and determine its key aspects, follow the steps below: ### 1. Simplifying the Function Start by simplifying the exponential expression: \[ f(x) = \frac{1}{2} \cdot 27^{\frac{2x}{3}} \] **Step 1:** Express 27 as a power of 3. \[ 27 = 3^3 \] **Step 2:** Substitute \( 27 \) with \( 3^3 \) in the function. \[ f(x) = \frac{1}{2} \cdot \left(3^3\right)^{\frac{2x}{3}} \] **Step 3:** Apply the power of a power property \((a^m)^n = a^{m \cdot n}\). \[ f(x) = \frac{1}{2} \cdot 3^{3 \cdot \frac{2x}{3}} = \frac{1}{2} \cdot 3^{2x} \] **Step 4:** Recognize that \( 3^{2x} \) can be written as \( (3^2)^x = 9^x \). \[ f(x) = \frac{1}{2} \cdot 9^x \] **Simplified Function:** \[ f(x) = \frac{1}{2} \cdot 9^x \] ### 2. Key Aspects of the Function #### **a. Initial Value** The initial value of the function is the value of \( f(x) \) when \( x = 0 \). \[ f(0) = \frac{1}{2} \cdot 9^0 = \frac{1}{2} \cdot 1 = \frac{1}{2} \] **Initial Value:** \( f(0) = \frac{1}{2} \) #### **b. Simplified Base** From the simplified function, the base of the exponential expression is: \[ 9 \] **Simplified Base:** \( 9 \) #### **c. Domain** The domain of an exponential function \( a^x \) is all real numbers, as you can raise a positive base to any real exponent. **Domain:** \( \mathbb{R} \) (All real numbers) #### **d. Range** The range of an exponential function \( a^x \) where \( a > 1 \) is all positive real numbers. Since the function is multiplied by \( \frac{1}{2} \), it scales the range but does not change its nature. \[ f(x) = \frac{1}{2} \cdot 9^x > 0 \quad \text{for all } x \in \mathbb{R} \] **Range:** \( (0, +\infty) \) ### **Summary** - **Initial Value:** \( f(0) = \frac{1}{2} \) - **Simplified Base:** \( 9 \) - **Domain:** All real numbers (\( \mathbb{R} \)) - **Range:** All positive real numbers (\( y > 0 \))