Why is y=sqrt(x) not reflected in the x-axis
Favorites|Homepage
Subscriptions | sitemap
HOME > > Why is y=sqrt(x) not reflected in the x-axis

Why is y=sqrt(x) not reflected in the x-axis

[From: ] [author: ] [Date: 12-04-22] [Hit: ]
there are ALWAYS two possibilities for the value of y (the positive or negative value).So if we can only graph functions (which is all your graphing calculator can do), then we CANNOT graph this inverse (y = ±√x) without graphing the TWO distinct functions:y₁ = √xy₂ = -√xEdit:I think the real answer to your question will be somewhat unsatisfying.The answer to your question (it seems) is a matter of convention:You state that: given x² = α² yields TWO solutions: x = ±α (of course unless α = 0, in which case theres only a single solution).The way to write such a solution is:x² = β --> x = β^½By convention,......
Surely y=sqrt(x) is equivalent to y=±x, and so there would be two values for x for every value of y? Why is this not represented in the graph for it?

-
This is a VERY good observation. This is an example where the inverse of a function is NOT a function:

y² = x --> y = ±√x

This is CLEARLY NOT a function because given a value, x, there are ALWAYS two possibilities for the value of y (the positive or negative value).

So if we can only graph functions (which is all your graphing calculator can do), then we CANNOT graph this inverse (y = ±√x) without graphing the TWO distinct functions:

y₁ = √x
y₂ = -√x

Edit:

I think the real answer to your question will be somewhat unsatisfying. The answer to your question (it seems) is a matter of convention:

You state that: given x² = α² yields TWO solutions: x = ±α (of course unless α = 0, in which case there's only a single solution). The way to write such a solution is:

x² = β --> x = β^½

By convention, when you write it in this form, it's UNDERSTOOD that β^½ represents ALL possible solutions (i.e. both positive and negative in this case).

However, we can define a class of functions for real roots, for instance for this:

β^½ --> √β

We DEFINE this function such that it returns the POSITIVE (and real) solution to β^½. Given this DEFINITION, we can list all of the solutions to β^½:

β^½ --> β = √β and β = -√β

Likewise we define "higher" roots:

β^⅓ --> ∛β <-- in this case we define the cube root as the ONLY real solution to β^⅓

β^¼ --> ∜β <-- in this case, there are always TWO real solutions, so we define the quartic root as the positive real solution to β^¼

-
Surely it is not... y = sqrt(x) would be equivalent to y^2 = x for positive values of x.

You can't take a square root of a negative number in the real numbers. That moves you outside the real numbers and into the complex numbers. So, in real numbers, y = sqrt(x) only makes sense for POSITIVE values of x, so you do not have any values of y (the output) for negative values of x.
12
keywords: axis,sqrt,not,is,Why,reflected,in,the,Why is y=sqrt(x) not reflected in the x-axis
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .