The confusion in the algebraic solution potentially lies in the fact the radical (which only has One principle real and positive solution) is converted to a polynomial (to transform the radical into a solvable algebraic expression) which creates a new expression with a minimum of Two Solutions.
@@MathCuriousity in a relation, for one 'x' you could possible get more than one values which are linked with 'x' example of a relation is equation of circle, (x² + y² = a²) this equation represents a circle with center of the circle at the origin and radius = a. Clearly, for one value of x, you will get two values of y. Hence, it is NOT a function for functions, for any value of x belonging in the domain of the function, you will only get a single value of y. for example, we have (y = ax) it's a straight line, with slope = a. clearly, for any value of x belonging to its domain (set of real numbers in this case) we only get one value of y which equals a times x. the reason why square roots are called functions: ( root (x) )² = |x| you see, when you square (root x) it equals |x| and not x. again, if you consider |x|, it is a function ==> for any value belonging to its domain, you will only get a single value of y so, for any value of x belonging to the domain of (root x) i.e. (set of all non negative real numbers) you only get one value of y hope that helps. feel free to ask if you didn't understand something!
Old math books are the best. No 'commercials' or other distractions from the content. Math teacher here, and also a math book collector. v
I'm enjoying these - especially since I'm studying to be a high school math teacher.
I wish you luck, but brace yourself for impact. Teaching highschoolers nowadays does not compare to what our teachers had to do.
Thank you!
The confusion in the algebraic solution potentially lies in the fact the radical (which only has One principle real and positive solution) is converted to a polynomial (to transform the radical into a solvable algebraic expression) which creates a new expression with a minimum of Two Solutions.
you shouldn't say the radical function is not a function because it doesn't pass the vertical line test. It is still a FUNCTION.
It’s still a RELATION. A function has a single consistent output for each input.
@@tamarleigh interesting! I apologize but what is the difference between a function, a relation, and an “operator” ?!
@@tamarleigh but what is the difference between a FUNCTION and an OPERATION?! Help me!
@@MathCuriousity
in a relation, for one 'x' you could possible get more than one values which are linked with 'x'
example of a relation is
equation of circle, (x² + y² = a²) this equation represents a circle with center of the circle at the origin and radius = a. Clearly, for one value of x, you will get two values of y. Hence, it is NOT a function
for functions, for any value of x belonging in the domain of the function, you will only get a single value of y.
for example, we have (y = ax) it's a straight line, with slope = a. clearly, for any value of x belonging to its domain (set of real numbers in this case) we only get one value of y which equals a times x.
the reason why square roots are called functions:
( root (x) )² = |x|
you see, when you square (root x) it equals |x| and not x.
again, if you consider |x|, it is a function ==> for any value belonging to its domain, you will only get a single value of y
so, for any value of x belonging to the domain of (root x) i.e. (set of all non negative real numbers) you only get one value of y
hope that helps. feel free to ask if you didn't understand something!