iven the functions f (x) = 4x−2 and g(x) = x2 +7x, find and simplify an expression
for the following, being sure to show your work:
(5.1) (f − g)(x)

To find the expression for \((f - g)(x)\), we need to subtract the function \(g(x)\) from the function \(f(x)\). Let's go step by step.

Given functions:
- \(f(x) = 4x - 2\)
- \(g(x) = x^2 + 7x\)

### Step 1: Set up the expression for \((f - g)(x)\)
\[
(f - g)(x) = f(x) - g(x)
\]

### Step 2: Substitute the given functions into the expression
\[
(f - g)(x) = (4x - 2) - (x^2 + 7x)
\]

### Step 3: Distribute the negative sign across the \(g(x)\) function
\[
(f - g)(x) = 4x - 2 - x^2 - 7x
\]

### Step 4: Combine like terms
- Combine the \(x\) terms: \(4x - 7x = -3x\)
- There are no other like terms to combine.

So, the simplified expression is:
\[
(f - g)(x) = -x^2 - 3x - 2
\]

This is the simplified expression for \((f - g)(x)\).

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the domain of the function \((f - g)(x)\)?
2. How would you find the zeros of the function \((f - g)(x)\)?
3. What is the derivative of \((f - g)(x)\) with respect to \(x\)?
4. How does the graph of \((f - g)(x)\) compare to the graphs of \(f(x)\) and \(g(x)\)?
5. Can you express \((f + g)(x)\) and simplify it?

**Tip:** When subtracting functions, always distribute the negative sign carefully to avoid mistakes in simplifying.

Given the functions f(x) = 4x - 2 and g(x) = x^2 + 7x, find and simplify an expression for (f - g)(x).

Learn how to subtract functions and simplify the expression for (f - g)(x) where f(x) = 4x - 2 and g(x) = x^2 + 7x. This detailed step-by-step solution covers function operations and polynomial subtraction. Suitable for students in Grades 8-10.

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