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), the transformations involve a horizontal shift, a vertical shift, and a vertical compression, which are essential for accurately graphing the new function based on the original f(x).
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Domain & Range of Transformed Functions
Horizontal Shifts
Horizontal shifts occur when the input variable x is altered by adding or subtracting a constant. In the function g(x) = -½ ƒ (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 and reflection involve scaling the output of a function and flipping it across the x-axis. In g(x), the coefficient -½ indicates a vertical compression by a factor of ½ and a reflection across the x-axis due to the negative sign. This means that the values of f(x) are halved and inverted, which significantly alters the shape and orientation of the graph.
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