Find the domain of $$ \mathrm{f}(\mathrm{x}) $$ $$ f(x)=\sqrt{6 x-42} $$

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UpStudy AI · Accepted Answer

To determine the **domain** of the function $$ f(x) = \sqrt{6x - 42} $$, we need to ensure that the expression inside the square root is **non-negative**. This is because the square root of a negative number is not a real number.

### Step-by-Step Solution:

1. **Set the Inside of the Square Root to Be Greater Than or Equal to Zero:**

$$
   6x - 42 \geq 0
   $$

2. **Solve for $$ x $$:**

$$
   6x \geq 42
   $$
   
   $$
   x \geq \frac{42}{6}
   $$
   
   $$
   x \geq 7
   $$

### Conclusion:

The function $$ f(x) = \sqrt{6x - 42} $$ is defined for all real numbers $$ x $$ that are **greater than or equal to 7**.

- **Domain in Interval Notation:**

$$
  \boxed{\, [\,7,\ \infty\,) \,}
  $$

- **Domain in Set Builder Notation:**

$$
  \{\, x \in \mathbb{R} \mid x \geq 7 \,\}
  $$

### Visualization:

On a number line, the domain starts at $$ x = 7 $$ and extends to positive infinity. Here's a simple representation:

```
----|==========================>
    7
```

All values of $$ x $$ to the right of 7 (including 7 itself) are included in the domain of $$ f(x) $$.
The domain of $$ f(x) = \sqrt{6x - 42} $$ is all real numbers $$ x $$ such that $$ x \geq 7 $$.

Find the domain of \( \mathrm{f}(\mathrm{x}) \) \[ f(x)=\sqrt{6 x-42} \]

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