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.

a. ƒ(g(-1))

Accepted Answer

Hello. Today, we're going to be using the given information to solve 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 to solve the composite function u of v of negative 3. Before we start solving for the composite function, let's go ahead and write down some given information. We are told that u of x is an even function. So u of x is even. We are also told that v is an odd function, so v is odd. So before we continue any further, 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 function and we want to input an integer whether it be positive or negative, the output value is going to be the same. So for example, let's say g of 2 is equal to 4. If we wanted to plug in the opposite value, which is g of negative 2, then the output value will remain the same based off of the definition of an even function. Now what about an odd function? Well an odd function states that f of negative x is equal to negative f of x. So if we were to plug in an integer, whether it be positive or negative into a function, the definition of the odd function states that the output values will be the opposite of each other. So for example, if we have g(2) equal to 4 and we wanted to find the opposite value which is g(-two) the odd function states that the value of g(-two) will be the opposite of g(positive2). So g(-two) will equal to -four. This is the definition for an odd function. So now that we know the definitions of an even and odd function, let's go ahead and start solving the composite function. So, in order to solve for the following composite function, the first thing we need to do is we need to solve for the innermost value, which in this case is going to be v of negative 3. If we take a look at the chart, we are not given the input value of negative 3, but we are given the input value of positive 3. If we were to plug in positive 3 into the v of x function, the output value will be negative 7. So we know that v of positive 3 is equal to negative 7. Now recall that v of x is an odd function. We want to solve for the opposite input value, which is negative 3. So if we were to use the definition of an odd function, that means that v of negative three will have the opposite output of v of positive 3. That is going to be negative negative 7, which will simplify to positive 7. That means v of negative 3 is equal to positive 7. And now that we know the value of v of negative 3, we can substitute v of negative 3 with positive 7 in the composite function. That will leave us with u of positive 7. Now we want to solve for the value of u of positive 7. If we take a look at the chart, we do have the input value positive 7. If we were to plug that in to the u of x function, the output value will be positive 9. So that means the value of u of positive 7 is equal to positive 9. That is going to be the final value for the composite function and that means the composite function u of v of negative 3 is equal to positive 9. This will be the solution to this problem. And that means the overall solution is going to be d. 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>

a. ƒ(g(-1))

Key Concepts

Even Functions

Odd Functions

Function Composition

Watch next