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?
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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
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
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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.
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.
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