Hello! I'd be happy to help you with that. Let's work through the composition of functions \( f(x) \) and \( g(x) \), denoted as \( (f \circ g)(x) \).
Given:
\( f(x) = \sqrt{2x + 18} \)
\( g(x) = x - 4 \)
To find \( (f \circ g)(x) \), we substitute \( g(x) \) into \( f(x) \):
\( f(g(x)) = f(x - 4) = \sqrt{2(x - 4) + 18} \)
Therefore, the composition \( (f \circ g)(x) = \sqrt{2x + 10} \).
Next, let's determine the domain of the composition function. The domain of a composite function is the set of all real numbers that make the composition function defined.
For \( f(g(x)) = \sqrt{2(x - 4) + 18} \) to be valid, the expression under the square root (\( 2(x - 4) + 18 \)) must be non-negative since you cannot take the square root of a negative number in real-valued functions.
Solving the inequality:
\( 2(x - 4) + 18 \geq 0 \)
\( 2x - 8 + 18 \geq 0 \)
\( 2x + 10 \geq 0 \)
\( 2x \geq -10 \)
\( x \geq -5 \)
Thus, the domain of the composite function \( (f \circ g)(x) = \sqrt{2x + 10} \) is \( x \geq -5 \). This means that all real numbers greater than or equal to -5 are included in the domain.
I hope this explanation helps! If you have any more questions or need further clarification, feel free to ask.
