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Absolute value functions are a fundamental concept in mathematics, transforming any input into its non-negative value. This transformation affects how the function behaves, especially around points where the expression inside the absolute value equals zero. Consider the function \( y = -2 + |x+1| \). Here, \(|x+1|\) signifies that whatever value of \(x+1\) is calculated, its absolute (always non-negative) form is used.
The key point for such a function is where the expression inside the absolute value turns zero: \( x + 1 = 0 \), which gives \( x = -1 \). This point divides the function into two cases: when \( x \geq -1 \) and when \( x < -1 \). Consequently, it creates a piecewise function that needs to be addressed separately over these intervals:

• For \( x \geq -1 \), the function becomes \( -2 + (x + 1) = -1 + x \).
• For \( x < -1 \), the function becomes \( -2 - (x + 1) = -3 - x \).

This behavior highlights the distinguishing feature of absolute value functions: their transformation at critical points changes how they are expressed, effectively splitting them into distinct linear functions.

X-intercepts in a graph occur where the function crosses the x-axis, meaning the output or \( y \) value is zero. To determine where these occur for the piecewise-defined function \( y = -2 + |x+1| \), we must solve each part separately:

• For \( y = -1 + x \), set \( y = 0 \) to find \( x \Rightarrow -1 + x = 0 \), which gives \( x = 1 \). Since \( x = 1 \) is within the interval \( x \geq -1 \), it is a valid intercept.
• For \( y = -3 - x \), set \( y = 0 \) to find \( x \Rightarrow -3 - x = 0 \), which gives \( x = -3 \). This fits the condition \( x < -1 \), determining this as an intercept.

Thus, the x-intercepts for this function are at \( x = 1 \) and \( x = -3 \). These points play a crucial role in helping understand where the graph crosses the x-axis, providing insights into the behavior of the function overall.

The y-intercept of a function is the point where it crosses the y-axis. This occurs where \( x = 0 \). To find it for the piecewise-defined function \( y = -2 + |x+1| \), substitute \( x = 0 \) into the applicable part of the function. Since \( 0 \geq -1 \), use \( y = -1 + x \):
Substituting gives \( y = -1 + 0 = -1 \).
Therefore, the y-intercept is \( (0, -1) \). This point is very useful in graphing as it gives a starting point on the y-axis for sketching the function's behavior and understanding its initial value when \( x = 0 \).

A function is said to have a discontinuity if there is a break or gap in its graph. For piecewise functions, discontinuity is often checked at the boundaries where each piece meets. In this case, we investigate at \( x = -1 \) for \( y = -2 + |x+1| \).
To determine discontinuity at \( x = -1 \), calculate the limit from both directions:

• From the left, using \( y = -3 - x \), the limit as \( x \rightarrow -1^- \) results in \( y = -2 \).
• From the right, using \( y = -1 + x \), the limit as \( x \rightarrow -1^+ \) also results in \( y = -2 \).

Since both limits are the same and equal the function's value at \( x = -1 \), there is no discontinuity here.
This confirmed continuity is important as it implies the function is smooth at \( x = -1 \), with no jumps or breaks.