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Chapter 2: Problem 24

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

Short Answer

Expert verified
The given equation \(x y - 5 y = 1\) does define y as a function of x, except when x = 5.

Step by step solution

01

Re-arrange the equation to solve for y

We start with the initial given equation \(x y - 5 y = 1\). In order to establish whether this is a function, we need to solve the given equation for y. This can be done by taking y common, followed by division by the coefficient of y, which gives the equation in the form \(y = \frac{1}{x-5}\).
02

Analyze the given equation

Next, analyze the obtained expression for y: \(y= \frac{1}{x-5}\). This equation implies that for any x, there exists a unique y value, except for x=5, where the denominator becomes 0, hence undefined.
03

Identify if the equation is a function

Since the equation we have now in step 2 - \(y = \frac{1}{x - 5}\) - gives a unique y-value for each x, except x=5, it can be stated that the equation does define y as a function of x, except at x = 5 where the function is undefined.

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Key Concepts

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

Function of x
In algebra, a function of x, often denoted as 'f(x)', is a relationship between two variables, x and y, such that for each input x, there is exactly one output y. Think of it as a machine: for every x that you put in, you get a single y out. In the original exercise, the equation provided is rearranged to isolate y, resulting in the function format, where y is written in terms of x:
This leads to the function expression $$y = \frac{1}{x - 5}$$.
With this function, you can input any value of x (except x=5, as this would make the denominator zero), and the function will output a corresponding y value, obeying the rule of a function of x.
Solving Algebraic Equations
The process of solving algebraic equations involves finding the value(s) of the variable(s) that make the equation true. In the case of the provided exercise, you are given an equation with two variables, x and y. The first step is to re-arrange the equation such that y is by itself on one side of the equation. Once the equation is in the form of y=f(x), you can then investigate its properties as a function of x. Solving equations becomes a crucial skill when understanding algebraic functions, as it allows you to express variables in a way that clarifies their relationships, like the dependence of y on x.
Domain of a Function
The domain of a function refers to all the possible input values (x values) that will produce a valid output (y values). For a function to be well-defined, every value in its domain must correspond to exactly one value in its range. In the original exercise, we have the function $$y = \frac{1}{x - 5}$$.
The domain is all real numbers except x=5 because when x equals 5, the function's denominator becomes zero, which is undefined in mathematics. The process of identifying the domain is essential whenever you're dealing with functions, to understand where the function can and cannot take values.
Algebraic Functions
Algebraic functions are functions that involve operations like addition, subtraction, multiplication, division, and taking roots of variables. They can be represented in a function notation where y is expressed explicitly in terms of x, often as f(x). In our exercise, the function $$y = \frac{1}{x - 5}$$
is an example of an algebraic function since it involves division鈥攁 rational operation. Understanding these functions involves identifying their domain, range, and any other unique characteristics (such as asymptotes and discontinuities) that can provide insight into how the function behaves graphically and numerically.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In graph of \((x-2)^{2}+(y+1)^{2}=16\) is my graph of \(x^{2}+y^{2}=16\) translated two units right and one unit down.

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+x+y-\frac{1}{2}=0$$

Express the given function \(h\) as \(a\) composition of two functions \(f\) and \(g\) so that \(h(x)=(f \circ g)(x)\). $$h(x)=\frac{1}{4 x+5}$$

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+y^{2}=25$$

Give an example of a circle鈥檚 equation in standard form. Describe how to find the center and radius for this circle.
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