Find the domain of the composite function . Please show all of your work.

f(x) = sqrt(x); g(x) = 2x + 12

You do not state what composite function you want.

Do you want f(g(x)) or g(f(x)) or f(x)*g(x) .....

Does the following reply answer your question?

http://www.jiskha.com/display.cgi?id=1307080009

sorry, f o g

f o g

= f(g(x))
= f(2x+12)
= √(2x+12)

Remember we cannot take the √ of a negative, so the domain is
2x + 12 ≥ 0
2x ≥ -12
x ≥ -6

To find the domain of the composite function, we need to determine which values of x are allowed as inputs for the composite function.

Step 1: Identify the domain of each individual function.

For f(x) = sqrt(x), the domain is all non-negative real numbers (x ≥ 0). This is because the square root function is defined only for non-negative values.

For g(x) = 2x + 12, there are no restrictions on the domain since it is a linear function. Therefore, the domain of g(x) is all real numbers.

Step 2: Determine the values of x that satisfy both functions.

To find the range of the composite function f(g(x)), we need to find the values of x that satisfy both the domain of f(x) and the output of g(x).

Since the domain of f(x) is x ≥ 0, we need to ensure that the output of g(x) is also non-negative.

Step 3: Set up the inequality.

We have g(x) = 2x + 12. To ensure a non-negative output, we set 2x + 12 ≥ 0.

Step 4: Solve the inequality.

2x + 12 ≥ 0
2x ≥ -12
x ≥ -6

Step 5: Determine the domain of the composite function.

The composite function f(g(x)) is defined only when both functions are defined. In this case, f(g(x)) is defined when x ≥ -6.

Therefore, the domain of the composite function f(g(x)) is x ≥ -6.