let $$  $$f(x)=x^2-1 and g(x)=2(x-1), find g(f(x))

Given the functions $ f(x) = x^2 - 1 $ and $ g(x) = 2(x - 1) $, we are tasked with finding $ g(f(x)) $.

### Step 1: Substitute $ f(x) $ into $ g(x) $.
We start by substituting $ f(x) $ into $ g(x) $. Since $ f(x) = x^2 - 1 $, we substitute this into $ g(x) $:

\[
g(f(x)) = 2(f(x) - 1)
\]

### Step 2: Replace $ f(x) $ with $ x^2 - 1 $.
Now, we substitute $ f(x) = x^2 - 1 $ into the expression:

\[
g(f(x)) = 2((x^2 - 1) - 1)
\]

### Step 3: Simplify the expression.
Simplify the terms inside the parentheses:

\[
g(f(x)) = 2(x^2 - 1 - 1) = 2(x^2 - 2)
\]

### Step 4: Final expression.
Distribute the $ 2 $ to get the final result:

\[
g(f(x)) = 2x^2 - 4
\]

Thus, the composition $ g(f(x)) $ is:

\[
g(f(x)) = 2x^2 - 4
\]

Would you like further details on the process?

Here are 5 related questions:
1. What is the domain of the composite function $ g(f(x)) $?
2. How would you find $ f(g(x)) $?
3. Can you find the inverse of $ g(x) $?
4. What would happen if the function $ g(x) $ was different, e.g. $ g(x) = 3(x-2) $?
5. How do you graph the composite function $ g(f(x)) $?

**Tip**: When working with function composition, always substitute carefully and simplify step by step to avoid mistakes!

let f(x)=x^2-1 and g(x)=2(x-1), find g(f(x))

Explore how to compute the composition of functions g(f(x)) where f(x) = x^2 - 1 and g(x) = 2(x - 1). This detailed solution covers function composition and algebraic simplification, making it ideal for students in Grades 9-12.

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