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. Common transformations include vertical shifts, horizontal shifts, reflections, and stretches or compressions. In the case of g(x) = -f(x) + 1, the negative sign indicates a reflection over the x-axis, while the '+1' indicates a vertical shift upwards by one unit.
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Domain & Range of Transformed Functions
Reflection
Reflection in mathematics involves flipping a graph over a specific axis. For the function g(x) = -f(x), the negative sign in front of f(x) indicates that the graph of f(x) is reflected over the x-axis. This means that all y-values of the original function f(x) are inverted, resulting in a graph that mirrors the original across the x-axis.
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Vertical Shift
A vertical shift occurs when a function is moved up or down on the Cartesian plane. In the function g(x) = -f(x) + 1, the '+1' indicates that after reflecting f(x) over the x-axis, the entire graph is then shifted upwards by one unit. This transformation affects the y-values of the function, adding one to each point on the reflected graph.
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