Function Transformations and Relations

This comprehensive page covers fundamental concepts of function transformations, relations, and function compositions. The content explores various mathematical operations and their applications in graphing and problem-solving.

Definition: A function is a special type of relation where each input has exactly one output, while a relation can have multiple outputs for a single input.

Highlight: The difference between relations and functions in math is crucial for understanding mathematical relationships.

Example: When finding inverse functions, swap x and y coordinates and solve for y. For instance, given f$x$ = -3x + 6, replace x with y and y with x, then solve for y to find the inverse.

Vocabulary:

• Domain: The set of all possible input $x$ values
• Range: The set of all possible output $y$ values
• Interval notation: A mathematical way to represent sets using brackets []

Definition: The step-by-step guide to function transformations includes:

1. Translations (vertical shifts using f$x$ ± k, horizontal shifts using f(x ± h))
2. Reflections (-f$x$ for x-axis reflection, f$-x$ for y-axis reflection)
3. Vertical dilations (stretches with factor > 1, compressions with 0 < factor < 1)

Highlight: For composition of functions, ensure the domain of the inner function matches the required input of the outer function.

Example: To verify if functions are inverses, check if f(g$x$) = x and g(f$x$) = x. For instance, verifying 3x-1 and 3x+3 are inverses requires substituting one into the other.