How do you know if something is a function algebraically?

Question

Like, without any vertical line test or plotting any points.

Can you explain simply? I'm not really sure if my answers are correct.
y³=x is a function
y=x³ is a function
x^4+y=0 is a function
-y+x^4=0 is a function
y=|x| is not a function
x=|y| is a function
y=2x+3 is a function (this I am sure is a function because this is a line)
y=8 is a function
x=-3 is a function

Answer 1

A function from A to B takes each point in A to a SINGLE point in B. A very simple definition, that's all you have to know. You haven't properly defined your function, i.e. which direction the mapping goes, so I am going to assume your mapping takes x to y.



y³=x is a function - Correct, y is a function of x, taking any x to its cube root

y=x³ is a function - Correct

x^4+y=0 is a function - Correct

-y+x^4=0 is a function - Correct

y=|x| is not a function - False. y is a function of x. Any x gets taken to |x|.

x=|y| is a function - False. Consider x = 1. There are TWO values of y which it maps to; -1 and 1, so it is not a proper mapping.

y=2x+3 is a function - Correct

y=8 is a function - Correct, anything gets taken to 8

x=-3 is a function - Correct, well, x is a function



In general, if you can rearrange it into the form y = ..., with no y on the right hand-side, then y is definitely a function of x. If you can't, then it will usually not be a function, but you still have to think about it.

Answer 2

The equation is a function if for each value of x, there is one and only one value for y



y^2 = x ... NOT a function

. . . if x = 1, y = 1 or y = -1



y = x^2 ... IS a function

. . . if x = 1 then y can only equal 1

. . . if x = -1 then y can only equal 1



Vertical line test for a function.



When you graph the equation, if a vertical line passes through the graph only once, the equation is a function. If any verticle line passes through the graph more than once, it is not a function.

Answer 3

To find if an expression is a function algebraically,you have to sub. in "-x" into the function.

You have 3 conditions



--IF it comes out the same as the original,then it is an even function eg.f(x)=x^2 ; f(-x)=(-x)^2 ; These to are the same so they are even functions.



--If it comes out as the opposite of the original,then its an odd function. eg. f(x)=x ; f(-x)=(-x) ; Opposite hence odd.



--If the end results are neither of the above statements,then its definitely neither of even and odd.



GRAPHICALLY



--You have to go with the VERTICAL LINE test.When it goes through at most one point,its a function.

If it goes through at least one point,its not a function.

Answer 4

anything of the form y=f(x) is an algebraic function



and all of the above can be plotted on the graph.





Any equation containing x and y can be plotted on a graph