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Chapter 11: Problem 2

Why does a set of points defined by a circle not satisfy the definition of a function?

Short Answer

Expert verified
A circle does not satisfy the definition of a function because each x-value can correspond to more than one y-value, failing the vertical line test.

Step by step solution

01

Understanding the Definition of a Function

A function is a relation between a set of inputs and a set of potential outputs such that each input is related to exactly one output.
02

Analyzing the Circle Equation

The equation of a circle with radius r and center at the origin is given by \(x^2 + y^2 = r^2\). This equation represents all points \(x, y\) that satisfy it.
03

Evaluating the Vertical Line Test

To check if a set of points represents a function, one can use the vertical line test. If any vertical line intersects the graph at more than one point, the graph is not a function.
04

Applying the Vertical Line Test to the Circle

If we draw a vertical line through any non-zero point on the x-axis that intersects the circle, it will intersect at two points (one above and one below the x-axis).
05

Conclusion

Since the vertical line intersects the circle at more than one point, each input x corresponds to more than one output y. Therefore, the set of points defined by a circle does not satisfy the definition of a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definition of a Function
A function is like a special kind of relationship between two sets: inputs and outputs. Imagine it as a cool machine where you put something in, and it gives you back exactly one thing. For every single input, there must be one and only one output.

This consistency is what makes functions reliable and predictable. You can think of it in simpler terms as any scenario where you always get the same result from the same input, such as a vending machine. If you press the button for 'chips', you always get chips and not an unexpected surprise.
Equation of a Circle
The equation of a circle is a neat way to describe all the points that make up the circle's outline. For a circle with its center at the origin (0,0) and radius r, the equation is: \[ x^2 + y^2 = r^2 \].

You can visualize this as the set of every possible (x, y) point that sits exactly r units away from the center.

It means if you pick any point (x, y) on this circle and plug it into the equation, it should hold true.
For example, if the radius (r) is 5, then any point satisfying \[ x^2 + y^2 = 25 \] will lie on the circle.
Vertical Line Test
The vertical line test is a straightforward way to see if a graph represents a function. Imagine drawing vertical lines (lines that go straight up and down) across your graph.

If any of those lines touch the graph at more than one point, then the graph does not define a function.

This is because it would mean a single input (x-value) maps to multiple outputs (y-values), violating the definition of a function.

For example, when applied to a standard circle, a vertical line will hit the circle at two points (one above the x-axis and one below), proving it's not a function.
Input and Output Values
In a function, every input value (x) has a unique output value (y). Inputs are also called domain elements, and outputs are called range elements.

Think of it as a coffee machine where selecting 'cappuccino' (input) always pours out a cappuccino (output). The consistency is crucial.

Using the circle example, for any given x-value within the circle's range, there are two possible y-values that satisfy the circle's equation \[ x^2 + y^2 = r^2 \].

This breaks the rule of having exactly one output for each input, which is why the point set of a circle does not form a function.

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Most popular questions from this chapter

Solve each system by the substitution method. $$ \begin{aligned} &x y=12\\\ &x+y=8 \end{aligned} $$

Solve each system by the substitution method. $$ \begin{aligned} &x^{2}+y^{2}=2\\\ &2 x+y=1 \end{aligned} $$

Solve each system by the substitution method. $$ \begin{aligned} &x y=-3\\\ &x+y=-2 \end{aligned} $$

Graph each circle. Identify the center if it is not at the origin. $$ x^{2}+y^{2}+8 x+2 y-8=0 $$

Graph each ellipse. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$
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