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how do you solve this Composition of functions

To help you solve the problem in the image, I'll analyze the content and walk you through the steps.

### Image Breakdown
The image contains a series of mathematical expressions involving composition of functions:

1. **\( g(x) = x - 5 \)**
2. **\( S = x - 5 \)**
3. **\( (x^2 - 7) + (S - x) \)**
4. **\( g(x) = x - 2x^2 \)**
5. **\( S - I = (g(x) - x^2) \)**

### Steps to Solve
The equations seem to involve the following:

1. **Function Composition**: We need to substitute one function into another or combine terms.
2. **Simplification**: Simplify the resulting expressions after substitution or combination.

#### Step 1: Understand the Functions
- \( g(x) = x - 5 \) is a simple linear function.
- \( S = x - 5 \) seems to define \( S \) in terms of \( x \).
- The expression \( (x^2 - 7) + (S - x) \) involves \( S \) defined earlier as \( x - 5 \).

#### Step 2: Simplify the Expressions
Let’s substitute \( S = x - 5 \) into the expression \( (x^2 - 7) + (S - x) \):

\[
(x^2 - 7) + ((x - 5) - x)
\]

Simplify \( S - x \):

\[
S - x = (x - 5) - x = -5
\]

So, the expression simplifies to:

\[
x^2 - 7 - 5 = x^2 - 12
\]

#### Step 3: Analyze the Other Equations
- \( g(x) = x - 2x^2 \) is another function.
- The expression \( S - I = (g(x) - x^2) \) can be evaluated by plugging in \( g(x) = x - 2x^2 \):

\[
S - I = (x - 2x^2 - x^2) = x - 3x^2
\]

### Conclusion
The primary steps involve substituting the definitions of \( g(x) \) and \( S \) into the equations and simplifying.

### Questions to Consider:
1. How do we generalize this method to other functions?
2. What is the significance of the function \( g(x) \) in this context?
3. Could the value of \( S \) change if \( x \) is a different function?
4. How does composition differ from simple substitution?
5. What are the implications if \( g(x) \) was a quadratic function instead?

### Tip:
Always verify the domain of the functions involved when performing composition, as this can affect the validity of your solutions.

Would you like to explore any of these questions in more detail?

Explore how to solve composition of functions problems step-by-step with detailed explanations. This guide covers function composition, linear functions, quadratic functions, and simplification techniques. Ideal for high school students and those studying advanced algebra concepts.

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