Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x.\) $$x=y^{2}+1$$

Step-by-Step Solution

Answer

No, the equation does not represent \(y\) as a function of \(x\) because it has two \(y\) values for one \(x\) value.

Step 1: Understand that a function correlates each x with a unique y

In a function, each value of \(x\) is associated to exactly one value of \(y\). This property excludes equations which yield multiple \(y\) values for the same \(x\). In such cases the equation does not represent a function.

Step 2: Identify y's correlation with x

In the provided equation, \(y\) is squared. This results in positive values irrespective of whether \(y\) is positive or negative. For any given value of \(y^2\), there are two possible values of \(y\) (one positive, one negative) and they would all result in the same \(x\) for given equation. So, an \(x\) can correspond to two different values of \(y\).

Step 3: Conclusion

Since, in the given equation, one \(x\) corresponds to more than one \(y\), it does not represent \(y\) as a function of \(x\). It means that, taking square root on both sides can give two values for \(y\) (one positive and one negative). Hence, \(y\) is not a function of \(x\).

Key Concepts

Relation Between Variables

In the world of mathematics, understanding the relation between two variables, such as \(x\) and \(y\), is crucial. To explore this relationship:

• Think of \(x\) as an input that undergoes a transformation to produce \(y\), the output.
• Equations or expressions often describe how \(x\) and \(y\) are connected.
• Instead of just numbers, this helps us understand patterns and predict outcomes.

In our exercise, the equation is \(x = y^2 + 1\). Here, \(x\) is explicitly expressed in terms of \(y\). Understanding how this equation affects the relation between \(x\) and \(y\) is the first step in determining if it is a function.

Function Definition

A function in mathematics is a special relation. It links each input, \(x\), to exactly one output, \(y\).

• This means, if \(x\) is your input, then you should be able to determine one distinct \(y\) as the result.
• The uniqueness of the output for each input is key.
• A quick visual check is the vertical line test on a graph; if a vertical line crosses the graph of the equation more than once at any location, then it is not a function.

In the equation \(x = y^2 + 1\), for any specific value of \(x\), finding more than one corresponding value for \(y\) would mean this relation does not fit the definition of a function. That's because it violates the principle of one-to-one mapping, necessary for a function.

Non-Function Example

Sometimes, an equation may look like it connects two variables nicely, but doesn't meet the criteria of a function. This happens when:

• An \(x\) value maps to more than one \(y\) value.
• This often occurs with equations involving squares, square roots, or absolute values.

Take \(x = y^2 + 1\):

• For a value like \(x = 2\), solving the equation \(2 = y^2 + 1\) means \(y^2 = 1\).
• This yields two possible solutions for \(y\): \(y = 1\) and \(y = -1\).

Having two \(y\) values for one \(x\) means it doesn't qualify as a function. Such examples are fundamental to understanding the boundaries and definitions in algebra and beyond.