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Chapter 1: Problem 14

For the following exercises, determine whether the relation represents \(y\) as a function of \(x .\) $$y=\frac{1}{x}$$

Short Answer

Expert verified
Yes, \( y = \frac{1}{x} \) represents y as a function of x.

Step by step solution

01

Understanding the Problem

To determine if a relation represents y as a function of x, we must check if each input value of x has exactly one output value of y associated with it.
02

Identify the Components of the Relation

The given relation is \( y = \frac{1}{x} \). Here, x is the independent variable (input) and y is the dependent variable (output).
03

Apply the Vertical Line Test

Plot the graph of the function \( y = \frac{1}{x} \). A relation is a function if no vertical line intersects the graph at more than one point.
04

Analyze the Graph

The graph of \( y = \frac{1}{x} \) is a hyperbola. When analyzing the hyperbola, no vertical line will intersect the graph at more than one point.
05

Conclusion

Since each value of x (except x = 0 as it is undefined) corresponds to exactly one value of y, \( y = \frac{1}{x} \) is a function of x.

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Key Concepts

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

Vertical Line Test
The vertical line test is an essential method to determine whether a graph represents a function of one variable. This test states that if a vertical line intersects the graph of a relation at more than one point, then the relation is not a function. To perform the test, imagine drawing vertical lines (parallel to the y-axis) through every possible x-coordinate on the graph.
  • If any vertical line crosses the graph more than once, the graph does not represent a function.
  • If each vertical line crosses the graph at most once, the relation is confirmed to be a function.
In our example with the function \( y = \frac{1}{x} \), plotting the graph shows that no vertical line intersects the hyperbola more than once. Thus, it satisfies the vertical line test, indicating it represents a function.
Hyperbola
A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations that satisfy a certain relation. The graph of the equation \( y = \frac{1}{x} \) creates a hyperbola. Here are some characteristics of the hyperbola in this context:
  • It has two separate branches, each located in one of the two opposite quadrants.
  • The branches get closer to the x-axis and y-axis as they extend but never actually touch these axes, indicating asymptotes, which are invisible lines the graph approaches.
Hyperbolas often occur in mathematical contexts involving fractions or ratios where variables cannot equal zero. In the graph of \( y = \frac{1}{x} \), the vertical line at \( x = 0 \) and the horizontal line at \( y = 0 \) are asymptotes.
Independent Variable
The independent variable in a function or relation is the input, and it often stands for an unknown quantity that is under control. In the equation \( y = \frac{1}{x} \), the independent variable is \( x \). The value of \( x \) is chosen freely, except it cannot be zero because division by zero is undefined.
  • The independent variable determines the output value by its presence in the function鈥檚 formula.
  • You can think of the independent variable as the value you change to see how it affects the other variable, \( y \).
Understanding the role of the independent variable helps in analyzing effects changes have on dependent variables.
Dependent Variable
The dependent variable represents the output in a function, dependent on the input or independent variable. In \( y = \frac{1}{x} \), \( y \) is the dependent variable, determined by the value of the independent variable \( x \). This implies:
  • The value of \( y \) changes in response to changes in \( x \).
  • It provides an output for specific input values assigned to \( x \).
In functions, the dependent variable relies on calculations involving the independent variable, forming an essential part of understanding relationships in algebra.

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Most popular questions from this chapter

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x)+5\)

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