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.

e. g(g(-1))

Accepted Answer

Hello. Today, we're going to be using the following information to solve for the following composite function. So we are told that the following table is given for some values of the functions u of x and v of x. This is where u of x is an even function and v of x is an odd function. We want to use this information on the table to solve for v of v of negative 3. Before we solve the composite function, let's just go ahead and write down some given information. So 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 and odd? Well, the definition of an even function states that if we have f of x, then the value of f of x will equal to the value of f of negative x. What this means is that if we were to plug in an integer, whether it be positive or negative, into a function, we will end up with the same result. So for example, let's say we have g of positive 2 and the value of g of positive 2 is equal to 4. Well, for the definition of an even function, if we were to plug in the opposite of 2, which is negative 2, then the output will remain the same and will still be positive 4. So this is the definition for an even function. Now, what about the definition of an odd function? Well, the definition of an odd function states that the value of f of negative x is equal to the opposite which is negative f of x. So using the function g of 2, which is equal to 4, if we were to plug in the opposite value, which is g of negative 2, the result based off of the odd function will be the opposite output, which is going to be negative 4. So this will be the definition of an odd function. So now that we've gone over what an even and odd function is, we want to use this information to solve for the composite function, v of v of negative 3. Now in order to solve for this composite function, we first want to start by solving for the innermost value of the composite function. The innermost value is defined by v of negative 3. So the first thing we're going to want to figure out is the value of v of negative 3. If we take a look at the table, the input values of x do not give us the output values of negative 3, but we do have an input value of positive 3. If we were to plug in the value of positive 3 into the v of x function, the output is going to be negative 7. So what we are given is that v of positive 3 is equal to negative 7. Now recall that v of x is an odd function. What we want is the opposite output of v of positive 3. It's v of negative 3. So if we input v of negative 3, which is the opposite value of the input given to us, then the output is going to be the opposite of v of positive 3. That is going to be negative negative 7, which simplifies to positive 7. So that means v of negative 3 is equal to positive 7 and now that we know this value we can substitute positive 7 for v of negative 3 in the composite function. That will leave us with v of positive 7. Now if we take a look at the chart, we do have the value of x for 7. If we were to plug in 7 into the v of x function, the output value is going to be negative 5. So based off of the chart, the final value for v of 7 is going to be negative 5. With that being said, that means the value of the composite function, which is v of v of negative 3, is going to equal to negative 5. This will be the answer to the problem. And with that being said, the overall answer is going to be b. 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>

e. g(g(-1))

Key Concepts

Even and Odd Functions

Function Composition

Evaluating Functions at Specific Points

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