Composition of even and odd functions from tables

Question

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions.

i. g(g(g(-1)))

Accepted Answer

Welcome back, everyone. In this problem, we're given a table with some values of u of x and v of x, where u of x is an even function and v of x is an odd function. And we want to find the value of v of v of v of negative 7. For our answer choices, a is 5, b is negative 5, c is 7, and d is negative 7. Now, what do we know here? Well, we're trying to find the value of v of v of negative 7. So that's a composite function. And recall that for a composite function, we evaluate the let's write that properly. Evaluate the innermost function first. So that means, to solve this problem, we're going to start by evaluating v of negative 7 and then the function after that that's inside and then the function that's left after that. What else do we know? Well, we also know the natures of or the nature of u of x and v of x. Recall that if a function is let's start with even. So if a function is even, that means if we put a negative value of f of x into f of x, it will be the same as the value of f of x. In other words, it's symmetrical, and we know that u of x is an even function. If a function is odd, then that means f of negative x would be equal to the negative value of f of x. In other words, it is not symmetrical. So while we work, let's keep that in mind as well. So if we're looking for v of v of v of negative 7, I'm gonna put that in a different color. That means we need to find the value of. V when X equals negative 7, but when we look on our list of choices, at no point is the value of X equal to negative 7. However, we know that v of x is an odd function. So what that means is that v of negative 7 would also be equal let me write that again in blue. Would also be equal to the negative value of v of 7. And we can find v of 7 because we have a value of x being equal to 7. Notice that when x is 7, v of x is negative 5. That means that that means that our function will now be v of v of negative negative 5. Because that's our value of v of 7. Negative negative 5 equals positive 5. And now we can evaluate v of positive 5. Notice that when x is 5, v of x equals 3. So that means v of 5 is actually equal to 3. Now finally, we can evaluate v of 3. When x is 3, v of x is negative 7. So, therefore, v of 3 equals negative 7. So that tells us then that v of v of v of negative 7 equals negative 7. And when we look back on our answer choices, that would have been answer choice d. Thanks a lot for watching everyone. I hope this video helped.

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Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

i. g(g(g(-1)))