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When discussing the domain and range of functions, we're essentially talking about the set of all possible input and output values, respectively. In the case of the absolute value function \( y = |x+4|-3 \), the domain is all real numbers, \( (-\infty, \infty) \). This is because the absolute value operation can accept any real number and will still remain a real number.

The range, however, is a little different. Since this particular function involves a shift, the smallest output value, or the bottom of our range, occurs at the vertex of the function. For \( y = |x+4|-3 \), the lowest y-value is -3, found at the vertex \((-4, -3)\). Consequently, the range of the function is \([-3, \infty)\). Thus, while the domain spans infinitely, the range will start at the lowest y-value and extend upwards.

Absolute value functions take any input and return the absolute value of it, meaning the non-negative version of that number. The general form is \( y = |x| \), and the graph of this function results in a V-shape, with the point of the V at the origin \(0, 0\).

When we have an expression like \( |x+4| \), we're shifting the basic graph of \( y = |x| \) horizontally. Specifically, the graph is moved to the left by 4 units. Adding numbers inside the absolute value symbol as \( |x+c| \) always moves the graph to the left by \( c \) units, whereas subtracting moves it right.

• For \( y = |x| \), the graph has a vertex at \((0, 0)\).
• In \( y = |x+4| \), the vertex shifts to \(( -4, 0)\).
• Any expression of the form \( |x-c| \) shifts right by \( c \) units.

Function transformations involve shifting or altering the basic graph of a function to achieve certain effects. With \( y = |x+4|-3 \), we're looking at both a horizontal shift and a vertical one. These transformations can help us easily identify the new vertex and shape of the graph.

Types of Transformations:

• Horizontal Shifts: This happens when you add or subtract a number inside the absolute value. Here, adding 4 inside \(|x+4|\) moves the graph 4 units left.
• Vertical Shifts: Adjusting the number outside moves the graph up or down. Here, subtracting 3 moves the entire graph down by 3 units.

These transformations influence not just how the graph looks, but also how the range and vertex are determined. The vertex's x-coordinate is affected by horizontal shifts, while the y-coordinate changes with vertical shifts. Hence, with \( y = |x+4|-3 \), the vertex calculated as \((-4, -3)\), shows these combined effects.