Problem 38 - Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) $$ y=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

Yes, \( y = \frac{1}{x^2} \) is a function of \( x \).

Step 1: Understanding Functions

A function is defined as a relation in which every input (or \(x\)) has exactly one output (or \(y\)). We need to verify if for each \(x\), there is only one corresponding \(y\) value.

Step 2: Analyzing the Given Equation

We are given the equation \(y = \frac{1}{x^2}\). This suggests that for any particular value of \(x\), \(y\) is determined as \(y = \frac{1}{x^2}\).

Step 3: Checking for Vertical Line Test

The vertical line test is a method to determine if a curve is a function by checking if any vertical line intersects the curve more than once. In this equation, substituting the same \(x\) always results in a single \(y\).

Step 4: Function Verification

Evaluate some values: - For \(x = 1\), \(y = \frac{1}{1^2} = 1\)- For \(x = 2\), \(y = \frac{1}{2^2} = \frac{1}{4}\)- For \(x = -2\), \(y = \frac{1}{(-2)^2} = \frac{1}{4}\)Each \(x\) value results in a unique \(y\), showing there is no ambiguity.

Step 5: Conclusion

Since each \(x\) value leads to one unique \(y\), the equation \(y = \frac{1}{x^2}\) does define \(y\) as a function of \(x\). There are no two different \(y\) values for the same \(x\), meaning it authentically represents a function.

Key Concepts

Vertical Line Test

The vertical line test is a quick visual way to determine if a graph represents a function. When you perform the test, you imagine drawing vertical lines (up and down) across the graph of the equation. If any vertical line touches the graph at more than one point, the graph does not represent a function. This is because multiple intersections indicate multiple outputs (or values of \( y \)) for a single input (or value of \( x \)).
In the case of the equation \( y = \frac{1}{x^2} \), each vertical line will intersect the curve only once, demonstrating that for each \( x \), there is a unique \( y \). Therefore, based on the vertical line test, this equation is indeed a function.

Unique Outputs

A key characteristic of functions is that each input corresponds to a single output. This means for every \( x \) there must be only one \( y \). If any \( x \) value gives multiple \( y \) values, then the relation is not a function.
Using the function \( y = \frac{1}{x^2} \), let's consider:

• For \( x = 1 \), we get \( y = \frac{1}{1^2} = 1 \)
• For \( x = 2 \), \( y = \frac{1}{2^2} = \frac{1}{4} \)
• For \( x = -2 \), \( y = \frac{1}{(-2)^2} = \frac{1}{4} \)

In each case, there is a distinct \( y \) for a given \( x \), proving the outputs are always unique for every input. Thus, \( y = \frac{1}{x^2} \) satisfies this fundamental criterion of functions.

Function Verification

Verifying a function involves checking if each input produces a unique output across the entire domain of the function. We already know a function must pass the vertical line test and have unique outputs for each input.
By evaluating several points for \( y = \frac{1}{x^2} \), you confirm the equation's consistency in producing one \( y \) value per \( x \). When \( x = 1 \), \( x = 2 \), and \( x = -2 \), we've seen distinct, unconflicted outputs. Each instance reassures us that the equation represents true function behavior. Hence, \( y = \frac{1}{x^2} \) is verified as a legitimate function of \( x \). This verification process ensures understanding and precision when dealing with equations involving functions.