Here are the essential concepts you must grasp in order to answer the question correctly.
Function Definition
A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if a relation defines y as a function of x, we check if any x-value is paired with more than one y-value. In the case of the equation x = y^4, we can see that for each positive x, there are two corresponding y-values (positive and negative), indicating that y is not a function of x.
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Domain and Range
The domain of a relation 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 relation x = y^4, the domain consists of all non-negative real numbers since y^4 is always non-negative. The range includes all real numbers, as y can take any real value, resulting in a non-negative x.
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Inverse Relations
An inverse relation switches the roles of the input and output. For the relation x = y^4, if we consider the inverse, we would express y in terms of x, leading to y = ±(x^(1/4)). This highlights that for each positive x, there are two corresponding y-values, reinforcing that y cannot be defined as a function of x due to the multiple outputs for a single input.
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