f(x)=
x 
2
 +6x−2
g, of, x, equals, 2, x, minus, 10
g(x)=
2x−10
Find: , f, of, g, of, x
Find: f(g(x))

We are given two functions:

1. \( f(x) = x^2 + 6x - 2 \)
2. \( g(x) = 2x - 10 \)

To find \( f(g(x)) \), we need to substitute \( g(x) \) into \( f(x) \).

### Step-by-step:

1. First, express \( f(g(x)) \):
   \[
   f(g(x)) = f(2x - 10)
   \]
   This means that wherever there is an \( x \) in \( f(x) \), we replace it with \( 2x - 10 \).

2. Now substitute \( 2x - 10 \) into \( f(x) = x^2 + 6x - 2 \):
   \[
   f(2x - 10) = (2x - 10)^2 + 6(2x - 10) - 2
   \]

3. Expand the terms:
   \[
   (2x - 10)^2 = (2x)^2 - 2(2x)(10) + (10)^2 = 4x^2 - 40x + 100
   \]
   \[
   6(2x - 10) = 12x - 60
   \]

4. Combine everything:
   \[
   f(2x - 10) = 4x^2 - 40x + 100 + 12x - 60 - 2
   \]

5. Simplify the expression:
   \[
   f(2x - 10) = 4x^2 - 28x + 38
   \]

Thus, \( f(g(x)) = 4x^2 - 28x + 38 \).

Would you like more details or have any questions about this? Here are 5 questions related to this process:

1. How do we simplify quadratic expressions like \( (2x - 10)^2 \)?
2. What is the importance of substituting one function into another in calculus?
3. Can we find the inverse of \( f(g(x)) \)?
4. What happens if we find \( g(f(x)) \) instead of \( f(g(x)) \)?
5. How can we graph \( f(g(x)) \) and interpret its behavior?

**Tip:** When substituting functions, ensure that you correctly handle each variable and expand carefully to avoid errors.

f(x) = x^2 + 6x - 2, g(x) = 2x - 10. Find: f(g(x))

Solve the function composition f(g(x)) where f(x) is a quadratic function and g(x) is a linear function. This detailed step-by-step guide walks through the substitution and expansion process, with examples and simplified results. Ideal for high school students (Grades 10-12).

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