Is 0=x \sec ^2 x-5 a Function? A Look at Plotting and Simplifying

In summary, the conversation discusses the equivalence of 0=\frac{cos^2x}{x}-\frac{1}{5} and 0=x \sec ^2 x-5, but notes that the two are different graphs as they are not functions and have different values for other x. The example of x2- 1= 0 and 1- x2= 0 is also used to illustrate this point.
  • #1
UrbanXrisis
1,196
1
[tex]0=\frac{cos^2x}{x}-\frac{1}{5}[/tex]

[tex]\frac{1}{5}=\frac{\cos x \cos x}{x}[/tex]

[tex]5=\frac{x}{\cos x \cos x}[/tex]

[tex]0=x \sec ^2 x-5[/tex]

is this true? when I plot this, it doesn't give me the same function
 
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  • #2
I'm pretty confident that is true, but this isn't a function. You can say that [tex]0=\frac{cos^2x}{x}-\frac{1}{5}[/tex] implies and is implied by [tex]0=x \sec ^2 x-5[/tex], but that does not mean the graph of [tex]y=\frac{cos^2x}{x}-\frac{1}{5}[/tex] will be the same as the graph of [tex]y=x \sec ^2 x-5[/tex]. The two graphs cross the x-axis at the same points, i.e. they have the same roots, but otherwise they are different.
 
  • #3
why are they different graphs?
 
  • #4
Why should they be? You did not start with a "function"- you started with an equation. Yes, it is true that [tex]0=\frac{cos^2x}{x}-\frac{1}{5}[/tex]
is equivalent to [tex]0=x \sec ^2 x-5[/tex], but if you write functions [tex]f(x)=x \sec ^2 x-5[/tex] and [tex]g(x)=\frac{cos^2x}{x}-\frac{1}{5}[/tex], then you are only saying f(x)= 0 and g(x)= 0 have the same roots. It says nothing about other values of x.

It is also true that x2- 1= 0 is equivalent to 1- x2= 0. Do think that means 1- x2= x2- 1 for all x?
 

Related to Is 0=x \sec ^2 x-5 a Function? A Look at Plotting and Simplifying

1. What is the purpose of plotting in math?

Plotting is a visual representation of mathematical data, functions, or relationships. It allows us to better understand and analyze the data or equations by seeing patterns and trends.

2. How do I plot a graph in math?

To plot a graph, you will need to have a set of coordinate axes (usually x and y) and a set of data points. Plot each data point on the respective axis, then connect them with a line or curve to represent the relationship between the data.

3. What is the importance of simplifying equations in math?

Simplifying equations is important because it helps us solve problems more efficiently and accurately. It also allows us to see the underlying relationships and patterns in the equation, making it easier to understand and manipulate.

4. How do I simplify an equation in math?

To simplify an equation, you can use properties of operations, such as combining like terms, distributing, and factoring. You can also use algebraic techniques, such as substitution or solving for a variable, to simplify an equation.

5. Can I use technology to plot and simplify equations?

Yes, there are many software programs and apps that can help with plotting and simplifying equations. Some common programs include Microsoft Excel, Wolfram Alpha, and Desmos. However, it is important to understand the concepts and methods behind the technology to effectively use it for math problems.

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