If f (x) = x² - 2x and g (x) = x + 1 find f g (x)

To find \( f(g(x)) \), we first need to evaluate \( g(x) \) and then substitute that result into \( f(x) \).

1. **Find \( g(x) \)**:
   \[
   g(x) = x + 1
   \]

2. **Substitute \( g(x) \) into \( f(x) \)**:
   \[
   f(g(x)) = f(x + 1)
   \]

3. **Evaluate \( f(x + 1) \)**:
   \[
   f(x) = x^2 - 2x
   \]
   Substitute \( x + 1 \) into \( f(x) \):
   \[
   f(x + 1) = (x + 1)^2 - 2(x + 1)
   \]

4. **Simplify**:
   \[
   = (x^2 + 2x + 1) - (2x + 2)
   \]
   \[
   = x^2 + 2x + 1 - 2x - 2
   \]
   \[
   = x^2 - 1
   \]

Thus, \( f(g(x)) = x^2 - 1 \).

If you need any further details or have questions, let me know! Here are some follow-up questions you might consider:

1. What is \( g(f(x)) \)?
2. How would you graph \( f(x) \) and \( g(x) \)?
3. Can you find the roots of \( f(g(x)) \)?
4. How would you find the derivative of \( f(g(x)) \)?
5. What would happen if \( g(x) \) was defined differently? 

**Tip**: Always remember to substitute carefully when working with composite functions!

Discover how to find f(g(x)) where f(x) = x² - 2x and g(x) = x + 1. This problem involves function composition, showcasing algebraic manipulation and simplification techniques. Ideal for Grades 9-11.

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