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.

c. ƒ(g(-3))

Accepted Answer

Hello. Today, we're going to be using the given information to solve for the following composite function. So we are told that the following table gives us some values for 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 table to help us find the value of the composite function, u of v of negative 5. So before we start solving for the composite function let's go ahead and write down some of the given information. The first thing we are told is that u of x is an even function. We are also told that v of x is an odd function. So before we move on, we need to ask ourselves, what does it mean for a function to be even or odd? Well, the definition of an even function states that if a function is even, then the value of f of x is equal to that of f of negative x. What this means is that if we plug in an integer into a function, whether it be positive or negative, the output will remain the same. So for example, let's say we have a function g and we plug in the value of 2. If we plug in 2, then let's suppose the output value is going to be 4. So based off of the definition of an even function, if we were to plug in the opposite value, which is g of negative 2, then the output value is going to be the same which is positive 4. So this is the definition for an even function. Now what about 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 you plug in an integer into a function, the opposite values which are positive and negative will always result in an opposite output. So for example, if we have g(2) equal to 4 and we plug in the value of negative two, the odd function states that the output for g(-two) will be the opposite of g(positive2). So g of negative 2 will equal to negative 4. This will be the definition of an odd function. So now that we understand the definitions of an even and odd function, let's go ahead and use this definition and the table to help us solve for the composite function. So we have 2 functions in the composite function. We have the v function plugged into u, but in order to start figuring out this value, we first want to find the innermost value which is going to be v of negative 5. So let's take a look at the given chart. Now the given chart does not give us the value of negative 5 in the input values, but what we do have is the opposite value which is positive 5. If we were to plug in positive 5 into the v of x function, the output value will be positive 3. So based off of the chart, we know that v of positive 5 is equal to positive 3. Now recall that v of x is an odd function and what we are looking for is the opposite input value which is going to be negative 5. So if we are looking for v of negative 5 based off of the definition of an odd function we will end up with the opposite output which is going to be negative 3. So v of negative 5 is equal to negative 3, and now that we know this value we can substitute v of negative 5 with negative 3 in the composite function. That will leave us with u of negative 3, and now we want to solve for the value of u of -three. If we take a look at the chart, we are not given the input value of -three, but we do have the input value of positive 3. If we were to plug in positive 3 into the u of x function, the output value will be positive 5. So what we do know is that u of positive 3 is equal to positive 5. And again, recall that u is an even function. What that means is that if we were to plug in the opposite input, which is u(-three), the output value will remain the same, which is going to be positive 5. So this is going to be the final value for the composite function and that means the composite function u of v of negative 5 is equal to positive 5. This is going to be the answer 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>

c. ƒ(g(-3))

Key Concepts

Even Functions

Odd Functions

Function Composition

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