Problem 64 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 9.) \(\sqrt{x}+y=12\)

Step-by-Step Solution

Answer

Yes, the equation defines y as a function of x.

Step 1: Isolate y in the equation

Given the equation \(\sqrt{x} + y = 12\). To determine if \(y\) is a function of \(x\), first isolate \(y\). Subtract \(\sqrt{x}\) from both sides: \(y = 12 - \sqrt{x}\).

Step 2: Apply the vertical line test concept

In order to verify if \(y\) is a function of \(x\), we must ensure that for each input \(x\), there is only one output \(y\). The expression \(y = 12 - \sqrt{x}\) is a real function because for every value of \(x\), there is exactly one corresponding value of \(y\).

Step 3: Conclusion

Since the expression \(y = 12 - \sqrt{x}\) ensures a single unique \(y\) value for each \(x\), the original equation \(\sqrt{x} + y = 12\) does indeed define \(y\) as a function of \(x\).

Key Concepts

Vertical Line Test

The vertical line test is a method used to determine whether a relation is a function. Imagine plotting a graph of your relation on a coordinate plane. If you can draw a vertical line anywhere on the graph, and this line intersects the graph at more than one point, then the relation is not a function. This is because a function can only have one output value for each input value.

This test is particularly useful because it provides a visual method to verify the uniqueness of outputs in a relation. In our example, when we consider the equation \( y = 12 - \sqrt{x} \), every value of \( x \) corresponds to exactly one value of \( y \). Hence, if you were to plot \( y = 12 - \sqrt{x} \), every vertical line would only cross the curve once.

By using this test, you can ensure your equations are properly representing a function, thereby confirming the mathematical relationship is valid.

Isolating Variables

Isolating a variable in an equation is a fundamental step in solving mathematical problems. It involves manipulating the equation to get one variable alone on one side of the equation. This process makes it easier to analyze and understand the relationship between variables.

In our exercise, to determine if \( y \) is a function of \( x \), you first isolate \( y \) in the equation \( \sqrt{x} + y = 12 \). This is done by subtracting \( \sqrt{x} \) from both sides of the equation, leading to \( y = 12 - \sqrt{x} \).

Isolating \( y \) makes it clear how changes in \( x \) affect \( y \). This step is essential for applying further tests or manipulations, like the vertical line test, because it shows the direct relationship between \( x \) and \( y \). Mastering variable isolation is crucial for both solving equations and performing advanced mathematics.

Real Functions

A real function is a type of function where both its domain (the set of input values) and range (the set of output values) are subsets of the real numbers. Real functions form the backbone of much mathematical analysis and understanding.

In our example, the isolated equation \( y = 12 - \sqrt{x} \) represents a real function. The domain of this function consists of all non-negative \( x \) values because the square root function \( \sqrt{x} \) is only defined for non-negative numbers. The range of the function would also be real, as \( y \) values are derived from real number operations.

Understanding real functions is key in mathematics, as they describe numerous real-world phenomena. Knowing how to work with them and recognizing them when they appear in equations further develops mathematical comprehension and problem-solving skills.