 In this video we provide the solution to question number 10 for the practice again number one for math 1050 We're given three functions f g and h and we're asked to identify which of these functions is odd Remember an odd function has symmetry with respect to the origin in particular You have f of negative x is equal to negative f of x that is the negative sign comes out when we evaluate the function At some negative expression there now for polynomials you can actually detect even an odd functions pretty quickly If a polynomial function has only odd powers like five and one that necessarily makes it odd But we can check that algebraically as well You have f of negative x that equals negative x to the fifth minus negative x right here for which you can buy Since they're odd powers the negative sign comes out you get negative x to the fifth again You have this negative negative x factoring out the negative sign like so you get x to the fifth minus x Which then equals negative f of x so that's gonna in general happen if you have a polynomial function with only odd powers It will be an odd function. That's actually where the name comes from so we should correctly Select choice f choice D is also a polynomial But now we have even powers x to the fourth minus one which one you can think of x to the zero zero is an even number We know it's not gonna be odd because there aren't only odd powers But this actually is an example of an even function We're not looking for that but it is in fact an even function I did want to point that out to see it you do the same thing you just look at g of negative x for which you place all The x's with a negative like so now even powers will absorb the negative sign negative one squared is in fact positive one So this becomes x to the fourth minus one Which is the same thing a g of x and that's what characterizes even functions a reflection across the y-axis That is replacing x with negative x doesn't actually change the formula doesn't change the graph now We're not looking for even functions. We're looking for odd functions So all we had to determine was that g was not odd the fact that's even is is not relevant for this question So we will remove g so detecting even odd functions with polynomials that as you just have some combinations Summer differences of powers of x that it may be some coefficients polynomials are very easy to detect It gets a little bit more tricky with other functions like this rational function here You notice you have an x to the first you have an x squared and on an x is zero down there You might be tempted to say it's not odd because there's an even power. That's actually false We're gonna see in a second that this is in fact an odd function You might be thinking it's not even because there's an odd powers again You have to be a little bit more careful with these functions The best thing to do is to actually check it replace x with negative x and try to simplify it So in the numerator you get a negative x and the denominator you get negative x squared plus one Now in the numerator since you have a negative x and that's the only thing there I could factor the negative one and put it in front of the whole fraction In the denominator a negative x squared gives us a positive x squared and then we didn't do anything to the one So you'll notice we have a negative x over x squared plus one that is just negative h of x And so we have h of negative x is equal to negative h of x that indicates that in fact h is likewise an odd function so we didn't include that in our answer as well So G is not odd because it was even F and H are both odd functions. So the correct answer would be choice E F and H are odd