Question

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)

Answer 1

To find \((f - g)(x)\), we need to subtract the function \(g(x)\) from \(f(x)\).

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

Now, we perform the subtraction:

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

Distributing the negative sign through \(g(x)\):

\[
= 5x^2 - 3x + 1 - 2x^2 - x + 2
\]

Now, combine like terms:

1. Combine the \(x^2\) terms:
\[
5x^2 - 2x^2 = 3x^2
\]

2. Combine the \(x\) terms:
\[
-3x - x = -4x
\]

3. Combine the constant terms:
\[
1 + 2 = 3
\]

Putting it all together:

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

### Domain Restriction:

Since both functions \(f(x)\) and \(g(x)\) are polynomials, they are defined for all real numbers. Therefore, the domain of \((f - g)(x)\) is:

\[
\text{Domain: } (-\infty, \infty)
\]

Thus, the final answer is:

\[
(f - g)(x) = 3x^2 - 4x + 3 \quad \text{with domain } (-\infty, \infty)
\]