If `f(x)=x-7 and g(x)=sqrt(x)`, what is the domain of `g@f`?

Options

A. `xle0`
B. `xge-7`
C. `xge0`
D. `xge7`

Explanation

To find the domain of the composite function \( g \circ f \), where \( f(x) = x - 7 \) and \( g(x) = \sqrt{x} \), we will follow these steps:

### Step 1: Define the composite function
The composite function \( g \circ f \) is defined as:
\[ g(f(x)) = g(x - 7) \]
Since \( g(x) = \sqrt{x} \), we can substitute \( f(x) \) into \( g \):
\[ g(f(x)) = \sqrt{f(x)} = \sqrt{x - 7} \]

### Step 2: Determine the conditions for the square root
The square root function \( \sqrt{x} \) is defined only for non-negative values. Therefore, for \( g(f(x)) \) to be defined, the expression inside the square root must be greater than or equal to zero:
\[ x - 7 \geq 0 \]

### Step 3: Solve the inequality
To solve the inequality \( x - 7 \geq 0 \), we add 7 to both sides:
\[ x \geq 7 \]

### Step 4: State the domain
The domain of the composite function \( g \circ f \) is all values of \( x \) that satisfy the inequality:
\[ x \geq 7 \]

Thus, the domain of \( g \circ f \) is:
\[ \text{Domain: } [7, \infty) \]

### Final Answer
The correct option is \( x \geq 7 \).

---