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The domain of a function is all the input values, the x-values, that the function can handle. In a piecewise function like \( g(x)=\left\{\begin{aligned}-1 & \text { if } x<0 \-x+2 & \text { if } x \geq 0 \end{aligned}\right.\), the domain covers all the x-values that each piece can accept. For the given function, there is the first rule, which is true for all x-values less than 0. The second rule holds for x-values starting from 0 and going up to infinity. When these conditions are taken together, the domain of the function encompasses all x-values because there is a specific rule for every x. This is why the domain is all real numbers, expressed as \((-\infty, \infty)\).The range of a function is all possible outputs or y-values. For this function:

• When \(x < 0\), the function yields a constant value of -1. So, -1 is part of the range.
• When \(x \geq 0\), the expression \(-x + 2\) decreases as x increases. Here, y can be 2 and smaller, touching every value down to \(-\infty\). Hence, the range of the piecewise function is \((-\infty, 2]\).

Understanding domain and range helps identify the scope of the function both for inputs and outputs.

Plotting piecewise functions requires understanding each piece separately, then combining them on one graph. The function \( g(x)=\left\{\begin{aligned}-1 & \text { if } x<0 \-x+2 & \text { if } x \geq 0 \end{aligned}\right.\) includes different rules depending on the x-value.Start by graphing the component for \(x < 0\). Here, g(x) = -1, which means it's a horizontal line across y=-1, stretching infinitely in the direction where x is less than 0.
Next, consider \(x \geq 0\). g(x) = -x + 2 is a linear equation. With a y-intercept of 2, or the point (0,2), and a slope of -1, it means that for every unit increase in x, y decreases by one unit. This graphed line angles downwards.Combining these:

• The horizontal line across y=-1 runs infinitely to the left from x=0.
• The line with slope -1 starts at (0,2) and descends to the right.

Bring the two sections together, realizing that there may be a sharp corner, or cusp, at the intersection point where x is 0.

A linear equation is at the heart of one part of this piecewise function. Specifically, the equation \( g(x) = -x + 2 \) is a straightforward example of a linear equation with a defined slope and y-intercept.The slope of the line is -1, meaning as x increases, y decreases. This is shown by the negative sign in front of the x. This indicates a downward trend or negative slope.The y-intercept is 2, showing where the line crosses the y-axis. In terms of graphing, any linear function appears as a straight line, which makes it simpler to graph once you know the slope and y-intercept.

• Slope (-1): Indicates the steepness and direction of the line.
• Y-intercept (2): The value of y where the line meets the y-axis.

Understanding these basics of linear equations allows you to quickly draft how a section of a piecewise function will behave and appear on a graph.