Problem 137 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x .\) $$x=5$$

Step-by-Step Solution

Answer

No, the equation does not represent \(y\) as a function of \(x\). This is because for the given \(x\) value, there are multiple possible \(y\) values.

Step 1: Understand the equation

The given equation is \(x = 5\). It states that the value of \(x\) is constantly \(5\), regardless of the \(y\) value. This specifies a vertical line in the Cartesian plane, where all points on this line have an \(x\) value of \(5\).

Step 2: Determine if y is a function of x

To analyze if \(y\) is a function of \(x\), we need to identify if each value of \(x\) corresponds to exactly one value of \(y\). In this case, the value of \(x\) is always \(5\), irrespective of the value of \(y\). Thus, for a single \(x\) value, there are multiple corresponding \(y\) values - all real numbers are possible \(y\) values. This means the equation does not define \(y\) as a function of \(x\) according to the definition of a function.

Key Concepts

Vertical Line Test

The vertical line test is a simple way to determine if a graph represents a function of x. If any vertical line intersects a graph at more than one point, the graph does not define y as a function of x.
Since a function implies a unique output (y-value) for each input (x-value), any vertical line crossing a graph in multiple places indicates multiple y-values for a single x-value.
Thus, in the case of the equation \(x = 5\), a vertical line at \(x = 5\) would intersect the graph at multiple points, confirming that y is not a function of x.

Cartesian Plane

The Cartesian plane is a two-dimensional coordinate system where each point is identified by an x-coordinate and a y-coordinate. This plane, defined by horizontal (x-axis) and vertical (y-axis) lines, is fundamental in plotting equations and understanding graphs.
For the equation \(x = 5\), the Cartesian plane shows a vertical line where every point along the line has the same x-value (5) but variable y-values. Each y-value corresponds to a different point on the vertical line, highlighting the diversity of outputs for the same x-input, again emphasizing the non-function nature of the graph.

Definition of a Function

A function in mathematics defines a particular relationship between two sets, typically between x (inputs) and y (outputs). According to the definition of a function, each element in the input set (x-values) must map to exactly one element in the output set (y-values).
In simpler terms, a function assigns a single y-value to each x-value. For the equation \(x = 5\), however, one x-value maps to many possible y-values. Therefore, it does not satisfy the requirement for y to be a function of x, since multiple results exist for the same input.

Real Numbers

Real numbers include all the numbers on the number line, encompassing all the possible values that y can take when x is fixed at a given value. They include rational numbers (such as fractions and integers) and irrational numbers (like \(\sqrt{2}\) and \(\pi\)).
In the case of the equation \(x = 5\), the y-values can be any real number. This vast range of possible y-values combined with a singular x-value (5) underscores why y is not a function of x, as one x leads to infinitely many y-values.