Graph the function $$ f(x)=6^{x}-3 $$. Give the domain and range. Which transformation is needed to graph the function $$ f(x)=6^{x}-3 $$ ? Choose the correct ar B. The graph of $$ f(x)=6^{x} $$ should be shifted up 3 units. C. The graph of $$ f(x)=6^{x} $$ should be shifted down 3 units. D. The graph of $$ f(x)=6^{x} $$ should be shifted to the right 3 units.

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To analyze the function $$ f(x) = 6^x - 3 $$, we can start by determining its domain and range.

### Domain
The function $$ f(x) = 6^x - 3 $$ is defined for all real numbers $$ x $$ because the exponential function $$ 6^x $$ is defined for all $$ x $$. Therefore, the domain is:
$$
\text{Domain: } (-\infty, \infty)
$$

### Range
The function $$ 6^x $$ takes on all positive values as $$ x $$ varies over the real numbers. Specifically, as $$ x $$ approaches $$ -\infty $$, $$ 6^x $$ approaches $$ 0 $$, and as $$ x $$ approaches $$ \infty $$, $$ 6^x $$ approaches $$ \infty $$.

Thus, the range of $$ 6^x $$ is $$ (0, \infty) $$. When we subtract $$ 3 $$ from $$ 6^x $$, the range shifts down by $$ 3 $$:
$$
\text{Range: } (0 - 3, \infty - 3) = (-3, \infty)
$$

### Transformations
The function $$ f(x) = 6^x - 3 $$ can be viewed as a transformation of the basic exponential function $$ g(x) = 6^x $$. Specifically, the transformation involves shifting the graph of $$ g(x) $$ down by $$ 3 $$ units.

Thus, the correct transformation is:
$$
\text{C. The graph of } f(x) = 6^x \text{ should be shifted down 3 units.}
$$

### Summary
- **Domain:** $$ (-\infty, \infty) $$
- **Range:** $$ (-3, \infty) $$
- **Transformation:** Shift down 3 units.
The graph of $$ f(x) = 6^x $$ should be shifted down 3 units.

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