Given 
f(x) = 2x2 + 1
 and 
g(x) = 7x − 9,
 find the following. f(g(x))


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

Given:
- \( f(x) = 2x^2 + 1 \)
- \( g(x) = 7x - 9 \)

### Step-by-step solution:

1. Start with \( f(g(x)) \). This means we substitute \( g(x) \) (which is \( 7x - 9 \)) into \( f(x) \):
   \[
   f(g(x)) = f(7x - 9)
   \]

2. Now, substitute \( 7x - 9 \) into the function \( f(x) = 2x^2 + 1 \):
   \[
   f(7x - 9) = 2(7x - 9)^2 + 1
   \]

3. Expand \( (7x - 9)^2 \):
   \[
   (7x - 9)^2 = (7x - 9)(7x - 9) = 49x^2 - 126x + 81
   \]

4. Now substitute this back into the function:
   \[
   f(7x - 9) = 2(49x^2 - 126x + 81) + 1
   \]

5. Distribute the 2:
   \[
   2(49x^2 - 126x + 81) = 98x^2 - 252x + 162
   \]

6. Add 1:
   \[
   f(g(x)) = 98x^2 - 252x + 162 + 1 = 98x^2 - 252x + 163
   \]

Thus, the final result is:
\[
f(g(x)) = 98x^2 - 252x + 163
\]

Would you like any further explanation or clarification?

### 5 Related Questions:
1. How do you find the inverse of a composite function like \( f(g(x)) \)?
2. What happens if you compute \( g(f(x)) \) instead of \( f(g(x)) \)?
3. How do you find the domain of the composite function \( f(g(x)) \)?
4. Can you explain how to expand and simplify binomials like \( (7x - 9)^2 \)?
5. How do you determine whether a composite function is one-to-one or onto?

### Tip: 
When composing functions, always carefully substitute the second function into the first and handle the algebra step-by-step to avoid mistakes in simplification.

Given f(x) = 2x^2 + 1 and g(x) = 7x − 9, find the following. f(g(x))

Discover how to solve the function composition problem f(g(x)) where f(x) = 2x^2 + 1 and g(x) = 7x - 9. This step-by-step solution involves key algebraic concepts such as polynomial expansion and function composition. Ideal for students in Grades 9-11.

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