Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of g, of, f, of, 0, .g(f(0)).
f, of, x, equals, 3, x, plus, 1
f(x)=
3x+1
g, of, x, equals, x, squared, minus, 2, x, minus, 6
g(x)=
x 
2
 −2x−6

To find the value of \( g(f(0)) \), we first need to evaluate \( f(0) \) and then substitute that result into \( g(x) \). The steps are as follows:

### Step 1: Find \( f(0) \)
The function \( f(x) \) is given by:
\[
f(x) = 3x + 1
\]
Substitute \( x = 0 \) into \( f(x) \):
\[
f(0) = 3(0) + 1 = 1
\]

### Step 2: Find \( g(f(0)) = g(1) \)
The function \( g(x) \) is given by:
\[
g(x) = x^2 - 2x - 6
\]
Now substitute \( x = 1 \) into \( g(x) \):
\[
g(1) = (1)^2 - 2(1) - 6 = 1 - 2 - 6 = -7
\]

### Final Answer:
\[
g(f(0)) = -7
\]

Would you like more details on any step? Here are five related questions for practice:

1. What is \( f(2) \) using the same function \( f(x) = 3x + 1 \)?
2. What is \( g(2) \) for the given function \( g(x) = x^2 - 2x - 6 \)?
3. What is \( g(f(2)) \)?
4. If \( h(x) = 2x + 3 \), what is \( h(f(1)) \)?
5. Can you find the value of \( f(g(1)) \)?

**Tip:** When dealing with composite functions like \( g(f(x)) \), always evaluate the inner function first before substituting its result into the outer function.

Given the definitions of f(x) = 3x + 1 and g(x) = x^2 - 2x - 6, find the value of g(f(0)).

Learn how to solve the composite function g(f(0)) using the functions f(x) = 3x + 1 and g(x) = x^2 - 2x - 6. This detailed step-by-step solution covers key algebra concepts, including working with composite functions and quadratic equations. Suitable for students in Grades 8-10.

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