Problem 26 - Algebra A Combined Function

Question

Determine whether each relation is also a function. $$ y=x^{2} $$

Step-by-Step Solution

Answer

Yes, \(y=x^2\) is a function as it maps each \(x\) to one unique \(y\).

Step 1: Understanding Functions and Relations

A relation is a set of ordered pairs. A function is a specific type of relation where every input (usually x) maps to exactly one output (usually y). To determine if a relation is a function, check if each input value is paired with exactly one output value.

Step 2: Analyzing the Given Relation

The relation provided is given by the equation \(y=x^2\). This equation represents a parabola opening upwards. It's important to analyze whether for each \(x\) value there is only one corresponding \(y\) value.

Step 3: Checking the Criteria for a Function

For \(y=x^2\), if you choose any real number for \(x\), there is exactly one output for \(y\) due to the squaring operation. For example, if \(x=2\), \(y=2^2=4\), and if \(x=-2\), \(y=(-2)^2=4\). Each distinct value of \(x\) results in only one value of \(y\), fulfilling the definition of a function.

Step 4: Conclusion

The equation \(y=x^2\) meets the criteria for a function since for every \(x\) there is exactly one corresponding \(y\). Therefore, \(y=x^2\) is a function.

Key Concepts

Parabola

When you see the equation \( y = x^2 \), you are looking at a parabola. A parabola is a specific type of curve, which is symmetric and U-shaped. In the equation \( y = x^2 \), this parabola opens upwards. This means as the value of \( x \) increases or decreases, the value of \( y \) will always increase.
Parabolas have distinct characteristics:

• The lowest or highest point is called the vertex. For the equation \( y = x^2 \), the vertex is at the origin (0, 0).
• The axis of symmetry is a vertical line that passes through the vertex. For \( y = x^2 \), this line is \( x = 0 \).
• They are mirror images on either side of this axis.

Parabolas occur in various real-world phenomena, like the shape of a satellite dish or the path thrown objects follow due to gravity. Understanding parabolas helps us recognize these patterns in different contexts.

Function Definition

A function is a special kind of relation. It relates an input to an output in a very specific way. The key rule for functions is that each input value (commonly identified as \( x \)) must map to exactly one output value (commonly \( y \)).
For example, the relation described by \( y = x^2 \) assigns a unique output \( y \) for each input \( x \). This consistency is what makes it a function.
Functions are fundamental in mathematics because they establish a predictable rule between variables. You'll often encounter functions not just in algebra, but also in calculus, sciences, and even computer science.
Here’s how you can check if a relation is a function:

• Inspect the inputs: Ensure that each input has one and only one output.
• Use the vertical line test: If no vertical line intersects the graph at more than one point, the graph represents a function.

Understanding functions helps to model real-world scenarios where one thing depends on another, like speed based on time or cost based on quantity.

Squaring Operation

The squaring operation is when a number \( x \) is multiplied by itself, represented as \( x^2 \). This operation is fundamental because it is a non-linear operation that transforms a number significantly.
In the context of the function \( y = x^2 \), squaring inputs results in non-negative outputs, as multiplying any number by itself, whether positive or negative, results in a positive number or zero.
Here are a few key points about the squaring operation:

• Values of \( x \) closer to zero result in smaller \( y \) values, and as \( x \) gets larger (positively or negatively), \( y \) increases substantially.
• The squaring operation is symmetric around zero. That means \( x \) and \(-x\) will yield the same \( y \) value, such as \( 2^2 = 4 \) and \((-2)^2 = 4 \).

Comprehending squaring is crucial because it frequently appears in algebra and geometry, especially in areas related to quadratic equations and assessing areas of squares.