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

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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 ($$ y > 0 $$)
- **Initial Value:** $$ f(0) = \frac{1}{2} $$ - **Simplified Base:** $$ 9 $$ - **Domain:** All real numbers ($$ \mathbb{R} $$) - **Range:** All positive real numbers ($$ y > 0 $$)

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

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