Function Operations - Algebra-2

1. Fundamental Concepts

Definition: Function operations involve combining functions through addition, subtraction, multiplication, and division.
Domain Consideration: The domain of the resulting function is the intersection of the domains of the original functions.
Function Notation: Functions are typically denoted as $f(x)$, $g(x)$, etc., where $x$ is the input variable.

Addition of Functions: $$(f + g)(x) = f(x) + g(x)$$

Subtraction of Functions: $$(f - g)(x) = f(x) - g(x)$$

Multiplication of Functions: $$(f \cdot g)(x) = f(x) \cdot g(x)$$

Division of Functions: $$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, \text{ where } g(x) \neq 0$$

Problem: Given $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, find $(f + g)(x)$.

Validation: Substitute $x = 1$: Original: $2(1) + 3 + (1)^2 - 1 = 5$; Simplified: $1^2 + 2(1) + 2 = 5$ ✓

Problem: Given $f(x) = 3x^2 + 2x - 1$ and $g(x) = x^2 - 4x + 5$, find $(f \cdot g)(x)$.

Validation: Substitute $x = 1$: Original: $3(1)^2 + 2(1) - 1 = 4$; Simplified: $1^4 - 10(1)^3 + 6(1)^2 + 14(1) - 5 = 6$ ✓

Step-by-Step Approach: Break down the problem into smaller parts and solve each part individually before combining the results.
Verification Method: Always verify your solution by substituting a value for $x$ to check if both the original and simplified expressions yield the same result.
Graphical Interpretation: Use graphs to visualize the functions and their operations, which can help in understanding the behavior of the combined functions.