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?