Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function. For a function f(x), its inverse f¯¹(x) satisfies the condition f(f¯¹(x)) = x for all x in the domain of f¯¹. In this case, since f(x) = x² - 4 is defined for x ≥ 0, we need to find f¯¹(x) by solving the equation y = x² - 4 for x.
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Domain and Range
The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For the function f(x) = x² - 4 with x ≥ 0, the domain is [0, ∞) and the range is [-4, ∞). The inverse function's domain and range will switch, meaning the domain of f¯¹ will be the range of f and vice versa.
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Graphing Functions
Graphing functions involves plotting points on a coordinate system to visualize their behavior. For f(x) = x² - 4, the graph is a parabola opening upwards, starting at the point (0, -4). When graphing the inverse function, it is essential to reflect the graph of f across the line y = x, which helps in understanding the relationship between the function and its inverse.
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