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.

g. ƒ (g(g(-2)))

Accepted Answer

Hello. Today, we're going to be solving for the given composite function using the following information. So we are told that the following table is given for some values of the functions u of x and v of x. We are told that u of x is an even function and v of x is an odd function, and we want to use this information to solve for u of v of v of negative 7. So before we start solving for the composite function, let's just write down some known information. First, we are told that u of x is an even function. We are also told that v of x is an odd function. So what does it mean for a function to be even or odd? Well, the definition of an even function states that f of x is equal to f of negative x. What this means is that if we have a general function and we decide to plug in an integer, whether it be positive or negative, we will end up with the same result. So for example, let's say we have a general function, g, and we decide to plug in the value of 4. And the output for g of 4 is going to be 5. Well, if we decide to plug in g of -four, which is the opposite of positive 4, the even function states that g of -four will also equal to 5. This is the definition for an even function. Now, what about the definition for an odd function? Well, the odd function states that f of negative x is equal to negative f of x. What this means is that if we plug in an integer into a function and decide to plug in a negative version of that integer, we will end up with the opposite result. So for example, if we have g(positive 4) and that is equal to 5, the odd function states that g(negative 4) will give us negative 5. So this is the definition for an odd function. We want to use these definitions and the table given to solve for the following composite function. So, notice that the composite function has 3 different functions we are dealing with. The innermost function is v of negative 7. The second most function on the outside is v of v of negative 7. And the outermost function is u of v of v of negative 7. In order to solve for this composite function, let's first start by finding the value of v of negative 7. Now the table given to us gives us the gives us the functions u of x and v of x, and it also gives us the values for x, which is the input values to the function. The other columns represent the outputs when the with the following corresponding x value. So we want to first start by solving for v of negative 7. Now if we take a look at the input values, we don't have negative 7, but we do have positive 7. If we were to plug in 7 into v of x, the resulting output will be negative 5. So we know that v of 7 is equal to negative 5. Now recall that v of x is an odd function and we've gone over the definition of an odd function. So if we wanted to input the opposite value of negative 7, which is v of negative 7, the result is going to be the opposite of negative 5. So we end up getting negative negative 5 which simplifies to positive 5. So what this means is that the value of v of negative 7 is equal to positive 5. Now that we know the value of v of negative 7, we're going to substitute v of negative 7 in the composite function with the value of 5. This will allow us to rewrite the composite function as u of v of positive 5. Now what we want to do is we want to solve for the innermost value of this composite function which is going to be v of 5. So if we take a look at the given chart, if we were to plug in the value of positive 5 into the v of x function, the output value will be positive 3. So based off the chart, we know that v of 5 is equal to positive 3. If we substitute in this value for the composite function, what we are left with is u of positive 3. So now what we want to do is we want to solve for the value of u of positive 3. And based off of the chart, if we were to plug in the value of 3 into the u of x function, the output value will be positive 5. So based off of the chart, we can say that u of positive 3 is equal to positive 5. Since this is the final value after simplifying the complex or the the function, that means that the composite function u of v of negative 7 is equal to positive 5. This is going to be the solution to the problem. And with that being said, the overall solution 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.

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>

g. ƒ (g(g(-2)))

Key Concepts

Even and Odd Functions

Function Composition

Evaluating Functions at Specific Points

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