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Chapter 1: Problem 24

Determine whether each equation defines \(y\) as a function of \(x .\) $$x y-5 y=1$$

Short Answer

Expert verified
Yes, the equation \(xy - 5y = 1\) defines \(y\) as a function of \(x\), for all \(x\neq 5\).

Step by step solution

01

Setting up the equation

First rewrite the equation \(xy - 5y = 1\) as \(y(x - 5) = 1\) by factoring out \(y\) from the left side.
02

Isolating y

To isolate \(y\), divide the equation \(y(x - 5) = 1\) through by \(x - 5\). This leads to \(y = \frac{1}{x - 5}\). It's clear that for each \(x\) there is exactly one value of \(y\).
03

Checking for ambiguities

Although it seems like this equation defines \(y\) as a function of \(x\), we must ensure that the denominator \(x - 5\) is not equal to zero, for this would make \(y\) undefined. However, \(x\) can take any value except 5, thus indeed \(y\) is defined against all values of \(x\) except 5, and the equation defines a function of \(x\).

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Key Concepts

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

Understanding Equations and Their Role
Equations are fundamental tools in algebra and mathematics, used to express relationships between variables. An equation works like a balance scale, where two expressions are set equal to each other. For instance, the equation \(xy - 5y = 1\) involves two variables, \(x\) and \(y\).
The aim is often to express one variable in terms of another, which can help delineate how changes in one affect the other. In this context, the equation depicts a linear relationship between \(x\) and \(y\), requiring manipulation to understand whether \(y\) can be defined explicitly as a function of \(x\).
  • An equation may represent a real-world problem or a purely mathematical scenario.
  • Solving an equation often involves simplifying or rearranging terms to isolate one variable.
In our exercise, simplifying \(xy - 5y = 1\) to \(y(x - 5) = 1\) helps in identifying potential functions.
Factoring: The Key to Simplification
Factoring is a crucial mathematical process used to make equations more manageable by breaking them down into simpler parts. It involves identifying common factors in an expression, such as numbers or variables, which can be factored out.
In the equation \(xy - 5y = 1\), factoring plays an essential role in transformation. By observing that \(y\) is a common factor in the terms \(xy\) and \(-5y\), we can factor it out to simplify the equation to \(y(x - 5) = 1\).
  • This step reveals more about the structure of the equation.
  • Allows for further operations, such as division, to isolate variables.
Factoring is an important tool in algebra as it simplifies solving, and sometimes it is needed for recognizing potential undefined points in functions.
Unveiling the Domain of a Function
The domain of a function includes all possible values of \(x\) for which the function is defined. It's vital to identify domains to avoid any undefined mathematical operations. For \(y = \frac{1}{x - 5}\), the function is undefined if the denominator becomes zero, as division by zero is not possible.
This calculation reveals that \(x = 5\) would make the function undefined. Hence, the domain of the function \(y\) in terms of \(x\) excludes the value \(x = 5\).
  • Determining the domain involves finding values for which the function remains valid and meaningful.
  • Domains help in understanding the limitations and behavior of the function over different values.
Identifying these constraints is crucial to correctly interpret and use a function in mathematical modeling and problem-solving.

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Most popular questions from this chapter

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}-2 x+y^{2}-15=0$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting about the \(x\) -axis, and then moving the resulting graph down four units.

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I must have made a mistake in finding the composite functions \(f \circ g\) and \(g \circ f,\) because I notice that \(f \circ g\) is not the same function as \(g \circ f\)

In Exercises \(67-70,\) graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned} x^{2}+y^{2} &=9 \\ x-y &=3 \end{aligned}$$
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