curve. 1. $$ f(x)=\frac{1}{7} \cdot 6^{x} $$

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To analyze the function $$ f(x) = \frac{1}{7} \cdot 6^{x} $$, we can explore several features of the function, such as its domain, range, intercepts, asymptotes, and behavior as $$ x $$ approaches positive and negative infinity.

### Step 1: Determine the Domain
The domain of the function $$ f(x) = \frac{1}{7} \cdot 6^{x} $$ is all real numbers, since the exponential function is defined for all $$ x $$.

### Step 2: Determine the Range
Since $$ 6^{x} $$ is always positive for all real $$ x $$, the function $$ f(x) $$ will also be positive. Therefore, the range of $$ f(x) $$ is $$ (0, \infty) $$.

### Step 3: Find the Intercept
To find the y-intercept, we evaluate $$ f(0) $$:
$$
f(0) = \frac{1}{7} \cdot 6^{0} = \frac{1}{7} \cdot 1 = \frac{1}{7}
$$
Thus, the y-intercept is at the point $$ (0, \frac{1}{7}) $$.

### Step 4: Asymptotes
The function does not have any vertical asymptotes since it is defined for all $$ x $$. As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches 0, which indicates a horizontal asymptote at $$ y = 0 $$.

### Step 5: Behavior as $$ x $$ Approaches Infinity
As $$ x $$ approaches positive infinity, $$ f(x) $$ approaches infinity since $$ 6^{x} $$ grows without bound.

### Step 6: Summary of Features
- **Domain**: $$ (-\infty, \infty) $$
- **Range**: $$ (0, \infty) $$
- **Y-intercept**: $$ (0, \frac{1}{7}) $$
- **Horizontal Asymptote**: $$ y = 0 $$
- **Behavior**: 
  - As $$ x \to -\infty $$, $$ f(x) \to 0 $$
  - As $$ x \to \infty $$, $$ f(x) \to \infty $$

If you need further analysis or specific calculations, please let me know!
The function $$ f(x) = \frac{1}{7} \cdot 6^{x} $$ is an exponential function with the following key features: - **Domain**: All real numbers. - **Range**: Positive real numbers greater than 0. - **Y-intercept**: At $$ (0, \frac{1}{7}) $$. - **Horizontal Asymptote**: $$ y = 0 $$. - **Behavior**: - As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches 0. - As $$ x $$ approaches positive infinity, $$ f(x) $$ approaches infinity. This function grows rapidly as $$ x $$ increases and is always positive.

curve. 1. \( f(x)=\frac{1}{7} \cdot 6^{x} \)

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