Given the functions, f(x) = 5x2 - 3x + 1 and g(x) = 2x2 + x - 2, perform the indicated operation. When applicable, state the domain restriction.

(f - g)(x)

To perform the operation \((f - g)(x)\), we subtract the function \(g(x)\) from \(f(x)\). Given:

- \(f(x) = 5x^2 - 3x + 1\)
- \(g(x) = 2x^2 + x - 2\)

We can compute \((f - g)(x)\) by subtracting the corresponding terms of the two functions:

\[
(f - g)(x) = (5x^2 - 3x + 1) - (2x^2 + x - 2)
\]

Now, distribute the negative sign to the terms in \(g(x)\):

\[
(f - g)(x) = 5x^2 - 3x + 1 - 2x^2 - x + 2
\]

Combine like terms:

\[
(f - g)(x) = (5x^2 - 2x^2) + (-3x - x) + (1 + 2)
\]

\[
(f - g)(x) = 3x^2 - 4x + 3
\]

### Domain
Since both \(f(x)\) and \(g(x)\) are polynomials, their domains are all real numbers \(\mathbb{R}\). Hence, the domain of \((f - g)(x)\) is also all real numbers:

\[
\text{Domain of } (f - g)(x) = \mathbb{R}
\]

### Final Answer:
\[
(f - g)(x) = 3x^2 - 4x + 3 \quad \text{with domain } \mathbb{R}
\]

Would you like further details or clarifications?

### Related questions:
1. What is \((f + g)(x)\) for the same functions?
2. How do we compute \((f \cdot g)(x)\) and simplify the result?
3. Can you solve for \(f(x) = g(x)\)?
4. What is the derivative of \((f - g)(x)\)?
5. How do you find the roots of \((f - g)(x) = 0\)?

### Tip:
Always check for domain restrictions when dealing with functions involving fractions, square roots, or logarithms.

Given the functions, f(x) = 5x^2 - 3x + 1 and g(x) = 2x^2 + x - 2, perform the indicated operation. When applicable, state the domain restriction.

(f - g)(x)

Learn how to subtract two quadratic polynomial functions f(x) = 5x^2 - 3x + 1 and g(x) = 2x^2 + x - 2, simplifying the expression and determining the domain. This example involves key algebraic operations such as combining like terms and polynomial identities. Suitable for high school students (Grades 9-12).

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