Problem 87 - College Algebra with Modeling and Visualization

Question

Determine if \(y\) is a function of \(x\). $$ x^{2}+y^{2}=70 $$

Step-by-Step Solution

Answer

No, \(y\) is not a function of \(x\) for the equation \(x^2 + y^2 = 70\).

Step 1: Identify the Equation Type

Examine the given equation \(x^2 + y^2 = 70\). This is an equation of a circle centered at the origin \((0, 0)\) with radius \(\sqrt{70}\).

Step 2: Understand the Definition of a Function

Recall that a relation is a function if every \(x\)-value is associated with exactly one \(y\)-value.

Step 3: Evaluate the Circle Equation

For a given \(x\)-value, substituting into \(x^2 + y^2 = 70\) usually yields two values for \(y\) (positive and negative square roots), which means \(y\) is not uniquely determined by \(x\).

Step 4: Apply the Vertical Line Test

Draw or imagine vertical lines passing through the circle. If any vertical line intersects the circle at more than one point, \(y\) is not a function of \(x\). The circle \(x^2 + y^2 = 70\) does not pass this test.

Step 5: Conclusion

Since there are \(x\)-values that correspond to more than one \(y\)-value, and the circle fails the vertical line test, \(y\) is not a function of \(x\).

Key Concepts

Vertical Line Test

The vertical line test is a simple visual way to determine if a curve or graph represents a function. Imagine drawing vertical lines (lines parallel to the y-axis) across a graph. If any of these lines touch the graph at more than one point, then the graph does not represent a function. This is because a function can only have one y-value for each x-value.For instance, consider our equation of a circle, given by the equation \( x^2 + y^2 = 70 \). If you were to draw this circle on a coordinate plane and pass a vertical line through different x-coordinates on the circle, you would frequently find two corresponding y-values. This means that the same x-value is paired with multiple y-values, violating the definition of a function.In summary:

• The vertical line test helps to determine the function status of a graph.
• If a vertical line intersects the graph more than once, it's not a function.
• The circle fails this test, indicating \( y \) is not a function of \( x \).

Circle Equations

Circle equations describe all the points that are equidistant from a central point. The general form of a circle's equation in a coordinate plane is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) represents the center of the circle, and \(r\) is the radius.In the given equation, \( x^2 + y^2 = 70 \), the center is at the origin \((0, 0)\). The radius is \( \sqrt{70} \), as derived from the equation form \( x^2 + y^2 = r^2 \). This form indicates that for any value of \( x \), there will typically be a positive and a negative value of \( y \) that satisfy the equation, hence multiple solutions.With circle equations:

• The fixed radius ensures that all points \((x, y)\) lie on the circle's edge.
• The circle is symmetric about both the x-axis and the y-axis.
• The simultaneous positive and negative roots of \( y \) imply it cannot define a function with respect to \( x \).

Function Definition

A fundamental concept in mathematics is the definition of a function. A function is a special type of relation. It is defined as a set of ordered pairs where each input, or \( x \)-value, is associated with exactly one output, or \( y \)-value.This distinction is crucial because it determines whether a relation can be reliably used to predict outputs based on inputs. In a typical function, knowing \( x \) means you can immediately find the corresponding \( y \), with no ambiguity.For the circle equation \( x^2 + y^2 = 70 \), the relation fails to meet this criterion. This is because for certain values of \( x \), the equation results in two possible values for \( y \). This violates the condition "one output for each input." Hence, the relation is not a function.To summarize:

• A function pairs each \( x \)-value with precisely one \( y \)-value.
• Mathematically, functions are predictable and non-ambiguous in output.
• The circle equation results in multiple \( y \)-values per \( x \)-value, hence it isn't a function.