explain why a curve that is symmetric about the x-axis is not the graph of a function?

Let me try to explain this without drawing anything. The most basic and simple answer is, it doesnt pass the verticle line test. If you were to draw a straight line vertically (up and down) it would intersect the graph at two points, which disqualifies it as a function.

Now here is the reasoning behind the vertical line test. We can relate a function to some kind of simple machine. lets say, an apple pie making machine. The way the apple pie making machine or APMM works, is that when you put apples in the machine, apple pies come out.

Now if we were to put apples in the machine and banana cream pies and apple pies came out, we would say that the machine doesnt work. as a matter of fact we would say the APMM was a piece of crap.

That is how functions work. When you put something in the "function machine" as an x value, you should only get one thing that comes out the other end of the "function machine" so only one y value. If we get two different y values for one single x value then the "function machine" is broken, and it isnt a function at all.

When the curve is symmetrical over the x axis, there are two y values for every x value, which fails the vertical line test and our "function machine example.

I hope this helps. keep trying to wrap your mind around it.

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You need to look at the graph of the equations. a) 2^x goes from near zero (in lower left of graph), passes through (0,1) then goes up to infinity as you go right, whereas y = 2^(-x) gives you a mirror image of that from left to right. Thus it is symmetric with respect to the y-axis b) logs of a base and the exponent functions (that base to the power x) are symmetric with respect to the line y = x, but the base has to be the same, so it can't be this choice c) y = log x is a curve and y = 10x is a line through the origin, so that eliminates this possibility d) meets the requirements I mentioned in b) so is indeed symmetric with respect to y=x e) not the choice, since the choice is d) A proof that y = b^x and logb x (b is the subscript) are symmetric with respect to y = x might be in order here. Rather than a proof, let's go for a demonstration. What this question is really asking is, are these functions the inverse of one another? Roughly speaking, if one function takes x to y, then the inverse takes y back to x. If x = 2, then y = 5^x gives you 25 so the point (2, 25) is in the graph of that y = 5^x. If you let x = 25, then y = log5 x is equal to y = log5 (25) and log5 (25) is equal to 2, so that the point (25.2) is in the graph of y = log5 x. So, these functions are inverses of each other. It can be seen that the graph of y = 5^x consists of all (x,y) from the graph of y = log5 x , except that the x-coordinate and y-coordinate are switched . That is what makes them symmetric with respect to the line y = x.

There are a bunch of different ways of stating it, but an equation is considered a function, that is the y term is written as f(x), IFF for every value the input has there is only ONE output, which is the same as saying that for any x there can be only one y,

So any second order equations of the from y squared = x is not a function.

It is clearer if you rewrite the equation as square root of y = x

If x is 0 then y is zero which is one to one, but if x is 1 y can be -1 or +1 if x is 2 y can be -4 or +4. For every value of x there are now two values of y, so by definition it is not a function.

You could make it a function by saying that the positive square root of y = x as one equation and the negative square root of y = x for another equation, which would cover the original equation domain but keep the one to one correspondence.

Symmetric To The X Axis

By definition, a function requires that there only be one y value for any given x value. The only way for a graph to meet that requirement and still have symmetry around the x-axis id for it to be the function y = 0.

Because there are two possible Y values for any given X value - one positive, and one negative. That is in direct opposition to the definition of a function, which says that for every X value, there is one and ONLY one Y value.

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