Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of left parenthesis, g, circle, f, right parenthesis, left parenthesis, 0, right parenthesis, .(g∘f)(0).
f, of, x, equals, 5, x, minus, 2
f(x)=
5x−2
g, of, x, equals, x, squared, minus, x, minus, 1
g(x)=
x 
2
 −x−1

To find the value of \((g \circ f)(0)\), we first need to understand what \((g \circ f)(x)\) means. This represents the composition of the functions \(g(x)\) and \(f(x)\), which means we first apply \(f(x)\), then apply \(g\) to the result of \(f(x)\).

Given:
- \(f(x) = 5x - 2\)
- \(g(x) = x^2 - x - 1\)

We are tasked with finding \((g \circ f)(0)\). This is done in two steps:

### Step 1: Find \(f(0)\)
First, substitute \(x = 0\) into \(f(x)\):
\[
f(0) = 5(0) - 2 = -2
\]

### Step 2: Find \(g(f(0)) = g(-2)\)
Now substitute \(f(0) = -2\) into the function \(g(x)\):
\[
g(-2) = (-2)^2 - (-2) - 1 = 4 + 2 - 1 = 5
\]

Thus, \((g \circ f)(0) = 5\).

### Final Answer:
\[
(g \circ f)(0) = 5
\]

Would you like more details or further explanation? 

Here are five related questions:
1. How would you solve \((f \circ g)(0)\)?
2. What is the general form of \((g \circ f)(x)\)?
3. How would you calculate \((g \circ f)(x)\) for any value of \(x\)?
4. What is the domain of \(g \circ f\)?
5. How do compositions of functions relate to inverse functions?

**Tip:** When dealing with function compositions, always evaluate the inner function first and then substitute its result into the outer function.

Given the definitions of f(x) = 5x - 2 and g(x) = x^2 - x - 1, find the value of (g∘f)(0).

Learn how to solve the function composition (g∘f)(0) where f(x) = 5x - 2 and g(x) = x^2 - x - 1. This step-by-step solution covers key algebraic concepts, including function composition and quadratic functions. Suitable for students in Grades 9-11.

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