Find f + g, f - g, fg, and f/g. f(x) = x² + x + 1, g(x) = x² -1

Verified step by step guidance

1. Identify the given functions: \(f(x) = x^2 + x + 1\) and \(g(x) = x^2 - 1\).
2. To find \(f + g\), add the two functions by combining like terms: write \(f(x) + g(x) = (x^2 + x + 1) + (x^2 - 1)\).
3. To find \(f - g\), subtract \(g(x)\) from \(f(x)\) by combining like terms: write \(f(x) - g(x) = (x^2 + x + 1) - (x^2 - 1)\).
4. To find the product \(fg\), multiply the two functions: write \(f(x) \cdot g(x) = (x^2 + x + 1)(x^2 - 1)\) and use the distributive property (FOIL) to expand.
5. To find the quotient \(\frac{f}{g}\), write \(\frac{f(x)}{g(x)} = \frac{x^2 + x + 1}{x^2 - 1}\) and note the domain restriction where \(g(x) \neq 0\) (i.e., \(x^2 - 1 \neq 0\)).

Key Concepts

Function Operations

Function operations involve combining two functions using addition, subtraction, multiplication, or division. For functions f and g, (f + g)(x) = f(x) + g(x), (f - g)(x) = f(x) - g(x), (fg)(x) = f(x) * g(x), and (f/g)(x) = f(x) / g(x), where g(x) ≠ 0.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents with coefficients. Here, f(x) and g(x) are quadratic polynomials, which means their operations result in new polynomials formed by combining like terms.

Domain Restrictions in Function Division

When dividing functions, the domain excludes values where the denominator function equals zero. For f/g, it is essential to find where g(x) = 0 and exclude those x-values to avoid undefined expressions.