Question

How do you decide whether the relation ${x^2} + {y^2} = 25$ defines a function?

Answer 1

In this question we have to find whether the relation defines the function or not. To proceed with the question we need to be clear about relation and function. A relation is basically a relationship between x and y coordinates, and function is its subset. A function is a type of relation in which each $x$ has a unique value of $y$.To check whether a relation defines a function or not, we need to be sure that for each value we input in $x$ it should give a unique value of $y$. Like for $x = 0$, if we have $y = \pm 5$, then this relation does not define a function.

Complete step by step solution:
We are given,
${x^2} + {y^2} = 25$
We can rewrite this equation as
$\Rightarrow {y^2} = 25 - {x^2}$
$\Rightarrow y = \sqrt {25 - {x^2}}$
For each value of $x$, there are two values of $y$.
For example,
For both $x = 5$ and $x = - 5$
Value of $y$ is $0$.

Note: There can be four types of relations- One-to-one, one-to-many, many-to-one, and many-to-many.
One-to-one – One value of $x$ has one value of $y$
one-to-many– One value of $x$ has many value of $y$
many-to-one– Multiple value of $x$ has one value of $y$
many-to-many– Multiple value of $x$ have multiple value of $y$
One-to-one and many-to-one relations define a function.
We can also check if a relation defines a function by “vertical line test”. In this test, you draw the graph of the equation and then draw a line parallel to $y$ axis, and if the line intersects the graph at more than two places, then it is not a function.