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.