Suppose that ƒ and g are both odd functions defined o...

Question

Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?

a. ƒg

b. ƒ³

c. ƒ(sin x)

d. g(sec x)

e. |g|

Accepted Answer

Hello. In this video, we are going to be determining whether each of the functions are even, odd, or neither. So, the problem says, to determine which of the following functions are even, or whether to determine that they are odd, we are given a note that F and G are both odd functions defined on the entire real line. Now, what's important about this problem is that we are told that F and G are both odd functions. In order for a function to be odd, they have the following property. If we plug in negative X into the function, their outputs are going to be the negative version of the original function. This is going to be important because we are going to need to use this characteristic on each of the functions that are given to us. So, let's go ahead and start with the first function, which is part one. Part one tells us that we have the function F of X plus G of X. Now, since both of these functions are odd, we can go ahead and use the identity and plug in negative X into each of the functions. That is going to give us F of negative X plus G of negative X. Now again, because we are told that they are odd functions, we can go ahead and rewrite them with a negative coefficient in front of the individual functions. So F of negative X will be rewritten as negative F of X, and the same can be applied to G of negative X. Since it is odd, it can be rewritten as negative G of X. Now, notice that we can factor out a -1 from both of the functions. Once we factor out a -1, that is going to allow us to rewrite the function as negative F of X plus G of X. Now notice that when we are given the original function F of X and G of X, the outputs of plugging in negative X give us the negative version of the original function. This satisfies the odd criteria, and that means that for part one, F of X plus G of X is going to be odd. Now we will go ahead and approach part two. Part 2 gives us the original function F of X squared. Now, we know that F of X is odd, so we will go ahead and plug in negative X. That is going to give us F of negative X squared. And again, by the odd criteria, we can rewrite this to be negative F of X squared. Now, we, in order to simplify the function, recall that any negative value that is raised to an even power is going to be is going to be positive, so we can simplify negative F of X squad to be positive F of X squared. Notice that the original function is given to us when we plug in the value of negative X. This satisfies the even criteria. In order for a function to be even. When we plug in the value of negative X, that is going to give us the original function of F of X. So, because we plugged in negative X and we got the original function, we can determine that the second part F squared is going to be even. Now let's go ahead and move on to the 3rd part. The third part gives us the function F of cosine of X. So we have F of cosine of X. We will go ahead and plug in negative X into our function, giving us F of cosine of negative X. Now recall the identity for cosine of negative X. Since cosine of negative X isn't even function, cosine of negative X can be rewritten as cosine of positive X. So we can go ahead and rewrite this function to be F. Of cosine of X. And notice that once we plug in the negative X into our function, we get the original function out. What this means is that the third function is going to be even. Now we will go ahead and approach the 4th function. The 4th function given to us is G of tangent of X. We will go ahead and plug in negative X into our function, giving us G of tangent of negative X. Now, what does tangent of negative X simplify to? While tangent of negative X. Can be rewritten as negative tangent of X. So, what we can go ahead and do is rewrite tangent of negative X as negative tangent of X, and we will be left with G of negative tangent of X. Now, because we know that G is an odd function, we can further rewrite the function with the negative coefficient in front of G, and this will give us negative G of tangent of X. This is the negative version of the original function G of tangent of X, and that means that the 4th function is going to be odd. Finally, let's go ahead and approach the 5th function, which is the absolute value of X. So we have the absolute value of F of X. We will go ahead and plug in negative X into this function, allowing us to rewrite the function as F of negative X. Since F of X is odd, we can further rewrite the function to be the absolute value of F of X. Now, the absolute value of any positive or negative number is going to be positive, so this will further simplify to be the absolute value of F of X. And since our output is the original function given to us, that means that the 5th function is going to be even. And that is how you determine the even or odd functions from the 5 functions given to us in this problem. And just to recap, the 1st function is on, the 2nd function is even, the 3rd function is even, the 4th function is on, and the 5th function is even, and that is going to be the solution to this problem. So I hope this video helps you in understanding how to determine even or odd functions, and I will go ahead and see you all in the next video.

Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?

a. ƒg
b. ƒ³
c. ƒ(sin x)
d. g(sec x)
e. |g|

Key Concepts

Odd Functions

Even Functions

Function Composition and Products

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