Suppose ƒ is an even function with

Question

Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))

Accepted Answer

Hello. Today, we're going to be determining the values of m of negative 3, m of n of 3, and m of m of negative 3. We are told that m of 3 is equal to 3 and m of 3 is an even function We are also told that n of 3 equal to negative 3 is an odd function So let's just go ahead and make a couple notes before we start finding the values. So if we take a look at the first given piece of information, we are told that the value m of 3 is equal to 3 and this is an even function. The definition of an even function states that f of negative x is equal to f of x. That means the negative value of the input to the function will give the same result as the positive value of the input of the function. Now if we take a look at the second piece of information, n(three) is equal to negative 3. Now again, we are told that n(three) is equal to negative 3 and this is an odd function. By the definition of an odd function, f of negative x is equal to negative f of x. What this means is that whenever you input a value into the function the output of the opposite value will always be the opposite of the given value. So let's go ahead and start by finding the first value which is M of negative 3. Now, since M of 3 is an even function, again, f of negative x will equal to f of x. That means that m of negative three will equal to the output of m of 3 which is equal to 3. So that means that the value m of negative three is equal to 3 and that is going to be our first solution. Now what we want to do is we want to find the second function, and that is going to be m of n of 3. In order to solve this value, we're first going to need to determine the value of n of 3. Now we already know what the value of n of 3 is going to be. That is going to be negative 3 because that is given to us. So that means if we plug in the value of negative 3 for n of 3, we are left with M of negative three. So what this means is we need to determine the value of M of negative three, but we've already done that in the first step. M of negative 3 is equal to positive 3 and that is going to be the same output for m of n of 3. So this is going to be our second solution. Now what we want to do is we want to find the value of the 3rd function, which is n of m of negative 3. So in order to enter this value, we're going to need to first determine the value of m of negative 3. Now in the previous two steps, we know that m of negative 3 is equal to positive 3. So if we plug this into the function, we are left with n of 3. Now what we need to do is we need to determine the value of n of 3, and we know that the value of n of 3 is given to us as negative 3. So what this means is that the value n of m of negative 3 is equal to negative 3, and that is going to be our final solution. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Suppose ƒ is an even function with ƒ(2) = 2 and g is an odd function with g(2) = -2. Evaluate ƒ(-2) , ƒ(g(2)), and g(ƒ(-2))

Key Concepts

Even Functions

Odd Functions

Function Composition

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