As seen in Example 3, the equation x^2+y^2=25 does not...

As seen in Example $3,$ the equation $x^{2}+y^{2}=25$ does not define $y$ as a function of $x$. Each graph in these exercises is a portion of the circle $x^{2}+y^{2}=25 .$ In each case, determine whether the graph defines $y$ as a function of $x,$ and if so, give a formula for $y$ in terms of $x .$

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Key Concepts

Expressing y in terms of x

Expressing y as a function of x involves rearranging an equation to isolate y on one side. This conversion is useful for understanding the relationship in a function format; however, not all equations can be rearranged to satisfy the definition of a function unless appropriate restrictions are applied.

Domain Restriction

Domain restriction involves limiting the set of x-values considered so that the relation meets the criteria of a function. By restricting the domain, a multi-valued relation (such as a circle) can sometimes be expressed as a single-valued function, ensuring that each x-value yields only one corresponding y-value.

Function

A function is a relation between a set of inputs and outputs where every input is associated with exactly one output. This fundamental concept ensures that when you choose any x-value from the domain, there is only one corresponding y-value, preventing ambiguity in the mapping.

Vertical Line Test

The vertical line test is a graphical method used to determine whether a curve represents a function. If any vertical line drawn through the graph intersects the curve in more than one point, the graph does not represent a function because it suggests that a single x-value corresponds to multiple y-values.

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