 In this video we present the solution to question number five for practice exam number one for math 12 10 And we're asked which of the following functions are odd Remember a function is odd if it's graph is symmetric with respect to the origin odd functions also have the property that f of negative x Is equal to negative f of x and so that's algebraic test is the litmus test We're going to use to decide if these functions are odd or not now do be aware that this question is asking for odd functions It could very well be swapped on your version of the test that will be asking for an even function Even functions are those functions which are symmetric with respect to the y-axis They have the algebraic property that f of negative x is equal to f of x now like I said on this question We're looking for odd functions, so we're just going to focus on the odd case But the way you test it's the same whether they ask you even or odd What you're going to compute is f of negative x if you can factor a negative sign out of everything That means the functions odd if the negative sign ultimately disappears then it was even and if neither of those two cases happens Then we'd say neither okay, so we're gonna have to test these one by one So it's three questions for the price of one so if we look at the first function f of negative x We're gonna end up with negative x to the fifth minus negative x like so for which if you take a Negative number to an odd power so like negative one to the fifth power. That's the same thing as just negative one Negative one to an even power will always be positive negative one to an odd power will always be negative And so this expression becomes negative x to the fifth we get plus x right here for which that is not f of x Right, so notice this is not an even function on the other hand if we factor out a negative sign This will give us x to the fifth minus x which x to the fifth minus x that is f of x This is negative f of x and so this is indicative that f is in fact an odd function So we want to keep that here If we look at g of x G of x right here. I guess we need to look at g of negative x We're going to get negative x to the fourth minus one Which like I said a moment ago if you take a negative number to an even power that actually will give you a positive So this becomes x to the fourth minus one which is the same thing as g of x This actually indicates to us that g is an even function. We're not looking for even functions So we're going to remove g of x from consideration So the final one is h of x here. We're going to look at h of negative x. This will give us negative x over negative x squared plus one For which on the top you just leave it as negative x on the bottom You have a negative x squared which will become a positive x squared and So notice you can actually factor the negative sign out in front of everything So you get x over x squared plus one and this then looks like negative h of x and this indicates to us that our function Is in fact odd So turned out f and h were odd functions, which then leads to the correct answer being e Now I want to be very careful on this one Like the reason we call these things even in odds is because for polynomials if all of the powers are odd It'll be an odd function if all of the powers are even that'll be an even function And that's where we get these names even in odd functions But that's only if we talk about polynomial functions if you talk about a rational function You'll notice that you have an odd power right here, and you have even powers right here There's a mix match right you have even powers and odd powers and this turned out to be an odd function because well The negative sign on the numerator stuck around all the negative signs in the denominator Disappeared and so the net effect was you had a factor of negative one in your side of your ratio So it gets a little bit more complicated Just using powers to recognize even in an odd symmetry and functions That's why it's best to use the algebraic test right here Just look at h of negative x or g of negative x and simplify the expression if you can do that This problem will be no problem for you whatsoever