Determine the domain and range of the function $$ f(x)=2 \sqrt[3]{108}^{2 x} $$. $$ \{x \mid $$ all real numbers $$ \} ;\{y \mid y0\} $$ $$ \{x \mid $$ all real numbers $$ \} ;\{y \mid y \geq 0\} $$ $$ \{x \mid x0\} ;\{y \mid $$ all real numbers $$ \} $$ $$ \{x \mid x \geq 0\} ;\{y \mid $$ all real numbers $$ \} $$

Question

Determine the domain and range of the function $$ f(x)=2 \sqrt[3]{108}^{2 x} $$. $$ \{x \mid $$ all real numbers $$ \} ;\{y \mid y>0\} $$ $$ \{x \mid $$ all real numbers $$ \} ;\{y \mid y \geq 0\} $$ $$ \{x \mid x>0\} ;\{y \mid $$ all real numbers $$ \} $$ $$ \{x \mid x \geq 0\} ;\{y \mid $$ all real numbers $$ \} $$

UpStudy AI · Accepted Answer

Let's analyze the function $$ f(x) = 2 \sqrt[3]{108}^{2x} $$ to determine its domain and range.

### Step 1: Simplify the Function
First, express the function in a more familiar form:
$$
f(x) = 2 \cdot \left(\sqrt[3]{108}\right)^{2x} = 2 \cdot 108^{\frac{2x}{3}} = 2 \cdot \left(108^{\frac{2}{3}}\right)^x
$$
Since $$ 108^{\frac{2}{3}} $$ is a positive real number, the function simplifies to:
$$
f(x) = 2 \cdot a^x \quad \text{where} \quad a = 108^{\frac{2}{3}} > 0
$$

### Step 2: Determine the Domain
The function $$ f(x) = 2 \cdot a^x $$ is an exponential function. Exponential functions are defined for all real numbers $$ x $$. Therefore, the domain is:
$$
\{ x \mid \text{all real numbers} \}
$$

### Step 3: Determine the Range
Since $$ a > 0 $$, the exponential term $$ a^x $$ is always positive for any real $$ x $$. Multiplying by 2 (a positive constant) preserves this positivity:
$$
f(x) > 0 \quad \text{for all real } x
$$
Thus, the range is:
$$
\{ y \mid y > 0 \}
$$

### Conclusion
The domain of the function is all real numbers, and the range consists of all positive real numbers.

**Answer:**

$$\{x \mid $$ all real numbers $$ \} ;\{y \mid y>0\} $$
The domain is all real numbers, and the range is all positive real numbers.

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