Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the changes made to the graph of a function based on modifications to its equation. These transformations include shifts, stretches, compressions, and reflections. In the given function g(x) = -(1/2)f(x+2), the graph of f(x) undergoes a horizontal shift, vertical compression, and reflection across the x-axis.
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Domain & Range of Transformed Functions
Horizontal Shifts
A horizontal shift occurs when the input of a function is altered by adding or subtracting a constant. In g(x) = -(1/2)f(x+2), the term (x+2) indicates a shift to the left by 2 units. This means that every point on the graph of f(x) will move leftward, affecting the overall position of the graph of g(x).
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Vertical Compression and Reflection
Vertical compression occurs when the output of a function is multiplied by a factor between 0 and 1, which reduces the height of the graph. In g(x) = -(1/2)f(x+2), the factor of -1/2 not only compresses the graph vertically but also reflects it across the x-axis. This means that the values of g(x) will be half of those of f(x) and inverted in sign.
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