in this video we're going to talk about how to tell if a function is even odd or neither we're going to talk about how to do it the easy way and also how to do it the way that your teacher wants you to do it so let's begin the first thing you need to know is how to tell if it's even a function is even if F of NE X is equal to F ofx so if you replace x with X and there's no change the new function that you get looks exactly like the original function then it's even now what about if it's odd it's odd if f ofx is equal toga F ofx so that is if you replace Negative X with X everything in the function every term has to change sign if one term changes sign and the rest do not it's not going to be odd now what about the last category when is a function neither even nor odd it's going to be neither if you plug in Negative X and if you do not get negative f ofx so what does that mean well let's say if you replace x with negative X and some signs change While others don't then it's going to be neither also if it doesn't equal F of X2 it's also neither so it can't equal negative f ofx or F ofx so basically as long as it's not even or odd it's neither so let's start with our first example let's say that f ofx is equal to x 4 + 3x^2 so is the function even odd or is it neither well here's how you do it the easy way look at the exponents is four even or odd four is an even number now what about two is it even or odd two is even so if all the exponents are even then the function is going to be even but now let's prove it let's show our work let's replace x with X so this is going to Bex raised to the 4th power + 3 * x^2 now what ISX raised to the 4 power that's basically x times itself four times x^2 ISX * itself 2 times two negatives is a positive three Nega will give you a negative result but four negatives will give you a positive result anytime you have an even number of negative signs it's going to produce a positive sign so this is going to be positive X 4th plus 3x^2 two negative signs will produce a positive result now notice that the function that we have is the same as the original function therefore this is equal to F ofx so we can make the statement that f ofx is equal to F ofx which is the definition of an even function now let's try our next example let's say that f ofx is equal to x to the 5th power + 2 x the 3 power is it even odd or neither now don't worry about the coefficient this is unimportant even though two is an even number that's not going to help us determine if it's even or odd look at the exponent five is it even or odd five is an odd number and three is also an odd number since all of the exponents are odd the function is going to be an odd function and now let's prove it let's replace x with negative X now whenever you have an odd number of negative signs the result will be negative for example x to the 3 powerx * X is POS x^2 time another Negative X that's X Cub so this is going to Bex to 5th power - 2x cub and all of that is equal to F ofx now what should we do in our next step in order to prove that this function is an odd function what you want to do is you want to factor out a Nega 1 if you take out 1X 5 /1 is postive x 5 all the signs will change -2X Cub will become posi 2x Cub now notice that this portion inside the brackets X 5th + 2x Cub is equal to the original function so at that point what you want to do is replace it with the original function f ofx so therefore we can say that f ofx is equal toga F ofx which is the definition of an odd function and that's how you can prove it now what about this one let's say that f ofx is equal to x^2 + 6 is it even or odd well you know X2 is an even component because it has an even exponent what about six well 6 is the same as 6 x to the 0 anything raised to the 0 power is one so X to Z is 1 which 6 * X to Z is 6 * 1 that's 6 and zero is an even exponent so the whole thing is going to be even so let's go ahead and prove it now let's replace x with negative X so this is going to be x^2 + 6 x * X is POS x^2 so we have x^2 + 6 notice that the function did not change so on the right side we can replace x^2 + 6 with f ofx on the left side we still have F of negx so whenever F of negx is equal to a positive F ofx then it's an even function so if you see a number it's even think of that number as being multiplied times x is z and Z is an even number like 2 4 6 what about X Cub - 8X is that even or is it odd well whenever you see an X it's basically x to the first power now one and three are odd numbers so therefore this is going to be an odd function now let's prove it so let's find find F ofx this ISX 3 power - 8 * xx^ the 3 power ISX Cub 8 * X is POS 8X so now notice that all signs change so to verify that it's odd take out a negative 1 if we factor out netive 1 is going to be positive X Cub - 8x and as we can clearly see X Cub - 8X is basically the same as F ofx so therefore f ofx is equal to F ofx which means that it's an odd function now what about this example x 3r - 5x^2 + 2 is it even or is it odd so notice that three is an odd exponent two is an even exponent whenever you see even and odd exponents together you know that it's going to be neither it's not even or odd so that's how you can tell when it's neither but let's prove it so let's plug in negative XX to the 3 power ISX Cub andx to the 2 power is POS x^2 so this is what we have now let's check to see if it's an even function if it's an even function right now the original function should be the same as a new function but notice that it's not the same the sign for X Cub changed but the sign for X2 and for two did not change so therefore it's not the same as the function so we can make the statement F ofx does not equal FX it's not even now to check to see if it's an odd function we need to take out or factor out negative 1 so all the signs will change Negative X Cub will become POS X Cub 5x^2 will become POS 5x^2 and 2 will become -2 now do we have the same function as the original function notice that these two are not the same X Cub looks very similar however NE 5x^2 is not the same as 5x^2 so therefore we can say that F ofx does not equal Nega F ofx which means that it's not an oddd function so so if it's not even and if it's not odd then by default it has to be neither so that's how you can prove if it's neither now let's spend a few moments talking about graphs an even function will be symmetric about the Y AIS an odd function is symmetric about the origin and if it's not symmetric about the origin or about the y- axis then it's neither so let's take a look at X2 because it has an even exponent we know it's an even function the graph of X2 looks like this it's basically a u let's do that again it's an upward u notice that there's symmetry about the Y AIS so that means that it's an even function now if you have a constant let's say like three that's an even function f ofx is the same as y by the way if you were to plot y equals 3am it's going to be a horizontal line at three and notice that this line is symmetric about the y- axis the left side looks exactly the same as the right side so therefore a constant by definition has even properties now what about the graph x Cub or Y y = XB we know it's an odd function this graph is an increasing function it looks like this as you can see there's symmetry about the origin this side in quadrant one looks exactly the same or it looks like a mirror image of the other side in quadrant 3 so that's an example of an odd function it's symmetric about the origin and then there's the graph f ofx = x or yal x to the first P one is an odd number but let's see why this function is odd using a graph so yal X is basically a line that increases at a 45° angle and as you can see it's symmetric about the origin the side in quadrant one looks like the same as the one in quadrant 3 so there symmetry about the origin which is the property of an odd function how would you describe a function that looks like this is it even odd or neither now here there's no symmetry about the X I mean about the Y AIS or the origin so this is neither now what about this one is it even odd or neither notice that the left side looks the same as the right side so therefore it's symmetric about the y- AIS which means that it's an even function here's another example for you determine if this one is even or if it's odd or if it's neither now it's not drawn perfectly to scale so use your good judgment so what would you say is it even odd or neither is it symmetrical about the Y AIS the origin or neither it's not symmetric about the y- axis the right side does not look the same as the left side however there is symmetry about the origin notice that quadrant 1 looks similar to quadrant 3 now this blue line could keep on going down so even though the height doesn't seem the same it can keep going in that direction and notice that Quadrant 4 looks like a reflection of quadrant 2 as you can see there symmetry about the origin which makes this function an odd function here's another one for you is this an even or is it an odd function so notice that the right side looks the same as the left side therefore it's an even function it's symmetric about the y- AIS what about this example is it even odd or is it neither well we know it's not even the right side does not look the same as the left side and it's not odd you can clearly see a difference between this section and this section it's not symmetric about the origin because quadrants uh four and quadrant 2 doesn't have the Symmetry about the origin they don't look the same so in this case this function would be uh neither it's neither even nor is it odd let's try one more example now what if we have let's say a circle let me draw better Circle it's not perfect but let's say it's it's a a well-rounded circle is it even odd or neither well the circle does have even properties as you can see the right side looks the same as the left side this whole side they're the same so there is symmetry about the Y AIS now what about about the origin is it um symmetric about the origin what would you say notice that the side in quadrant one looks like a reflection as the one in quadrant 3 so there's some symmetry about the origin and quadron 2 and four are symmetrical about the origin so this graph is symmetrical about the y axis and about the origin as well so that is does that make it even or odd now technically speaking we can't really say even or odd because it's not a function this function doesn't pass the vertical line test so we can't say it's an odd function it's not an even function maybe it's neither because it's not a function so think about that one