Graph each piecewise-defined function and state its domain...

Graph each piecewise-defined function and state its domain and range. Use transformations of the toolbox functions where possible.
$$g(x)=\left\{\begin{array}{ll}-(x-1)^{2}+5 & -2 \leq x \leq 4 \\2 x-12 & x>4\end{array}\right.$$

Key Concepts

Linear Functions

Linear functions are first-degree functions characterized by a constant rate of change, or slope, and a y-intercept. They graph as straight lines. Recognizing a linear function and understanding its slope and intercept are essential for quickly graphing its behavior and for determining its contribution to the overall behavior of piecewise-defined functions.

Quadratic Functions

Quadratic functions are polynomial functions of degree two and graph as parabolas. They are often expressed in vertex form, which clearly shows the position of the vertex and how the parabola is shifted and reflected relative to the standard quadratic function. Understanding the transformations of quadratic functions is key to graphing them efficiently.

Piecewise-defined Functions

Piecewise-defined functions are functions that have different expressions or rules over different parts of their domains. They are defined by specifying separate function rules for different intervals of the input variable, and analyzing them involves handling each piece individually while considering how they join together at boundary points.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined, and the range is the set of all possible output values. When working with piecewise-defined functions, it is important to determine the domain and range for each individual piece and then combine them appropriately, taking into account any endpoints or gaps.

Function Transformations

Function transformations involve shifting, reflecting, stretching, or compressing a basic function to form a new function. This concept allows one to graph a function by starting with a known base function, then applying horizontal or vertical shifts, reflections about the axes, and scalings accordingly. Recognizing these transformations simplifies graphing and analysis.

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