Question

Determine whether the equation defines y as a function of \(x .\) \(x=y^{2}\)

Short answer

No, the equation does not define y as a function of x.

Step-by-step solution

Understand the Definition of a Function

A function implies that for each input value of the independent variable (here, x), there should be exactly one output value of the dependent variable (here, y).

Rewrite the Given Equation

Consider the equation given: \[ x = y^2 \]Rewrite it in terms of y: \[ y = \text{±}\rk{x} \]

Check for Single Output for Each Input

From the rewritten equation, \[ y = \text{±}\rk{x} \] there are two possible values of y for each positive x: one positive \( \rk{x} \) and one negative \( -\rk{x} \).

Conclusion

Since each positive x value yields two possible y values, the equation \( x = y^2 \) does not define y as a function of x.

Key concepts

Functions and Relations

A relation in mathematics is a collection of ordered pairs where the first element comes from one set (the domain) and the second comes from another set (the range). Not all relations are functions.
A function is a special type of relation that links every input to exactly one output.
This unique pairing means that every time you plug in a value (called the independent variable), you get back exactly one result (called the dependent variable).
In our exercise, we are asked whether the equation \[ x = y^2 \] defines y as a function of x. This means we need to verify if each value of x is linked to only one value of y.

Dependent and Independent Variables

Understanding dependent and independent variables is crucial when studying functions.
The independent variable is what you input into the function. Typically represented by x, it can stand for any value from the domain.
The dependent variable is what you get out of the function. Typically represented by y, it depends on the value of x. For instance, when x = 1, y will yield one specific value in a function.
In the given exercise, if we consider \[ x = y^2 \], x is the independent variable and y is the dependent variable.

Function Definition

A function is a relation where every input is related to exactly one output.
In mathematical terms, a function f from a set X (the domain) to a set Y (the range) is a rule that assigns to each element x in X exactly one element y in Y, often written as \[ y = f(x) \].
To determine if \[ x = y^2 \] is a function, we can rewrite it in terms of y: \[ y = \text{±}\rk{x} \]
This results in two possible y-values for each positive x, thus violating the definition of a function.
Hence, \[ x = y^2 \] is not a function because it does not yield a single, unique output for each input.