WEBVTT
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Just as we can do operations with real
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numbers, we can do operations with functions.
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Let's look at a particular operation of
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functions called composition.
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Given two functions, f and g, the composite
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function f composed with g is defined by this.
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We read it f composed with g of x equals f of g of x.
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The domain of the composite function is the
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set of all numbers x in the domain of g such that g of x is in the domain of f.
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Let's look at an example.
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So, let's look at two functions: the first
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one, f of x equals 3 x is a linear function,
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and g of x equals 4 x squared plus 6 is a quadratic function.
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The first thing we want to do is find the
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composition of f with g at a particular
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value of x when x is 4.
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We'll see how this works.
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We would like to find f composed with g at 4.
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By the definition of composition, this is f
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of g of 4, and let's think about what we need to do.
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Here we're asked to find g of 4.
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Let's do that here.
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We know how to do this: g of 4 means when x
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is 4 find the value of the function g.
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So that's going to be 4 times 4 squared plus 6.
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So, we have 16 times 4, 64, plus 6, 70.
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We want to replace g of 4 with 70.
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So now we're asked to find the f of 70.
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F of 70 means to find the value of the
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function f when x is 70, so let's look at f of x.
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We want that x to be 70, and that is 210.
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In the second part of this example, we're
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asked to find g composed with g when x is negative 1.
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So, we start by writing down what we're
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asked to do: g composed with g of negative 1.
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By definition this means to find g of g of negative 1.
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Alright, so we need to work on the inside
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part and find g of negative 1.
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This means find the value of the function g when x is negative 1.
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So, we look up there.
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It's going to be 4 times negative 1 squared plus 6.
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Be careful here when you square a negative
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1, it's going to be positive 1, multiplied
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by 4 is 4, and then add the 4 to the 6.
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So, g of negative one is 10, and now we're
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asked to find g of 10 or the value of the function g when x is 10.
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You can look up here: we want to find the
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value of this function when x is 10.
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So, 10 squared is 100, times 4 gives us 400,
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add 6 to that and we get our answer of 406.
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In this next part, we don't have a
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particular value of x in mind.
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We'd like to find a rule for the
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composition of these two functions.
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So, let's start by finding, start by using
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the definition: g composed with f of x is g of f of x.
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Alright, so we know that f of x is 3 x, so
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let's make the substitution, and we see that
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what we want is to find g of 3 x.
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So here I'm going to replace the x with 3 x,
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very carefully paying attention to what's
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inside the parentheses and what's going to be squared.
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Alright, I need to do the exponent first so
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this is 4 times 9 x squared plus 6, and if I
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multiply here, I'll get 36 x squared plus 6.
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We're also asked to state the domain of this
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function, so let's start at the beginning.
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You want to take any x in the domain of f: f
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is a linear function so its domain is all real numbers.
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This is going to give us an output of a
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function value that is still going to be a
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real number because the range of any linear
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function is all real numbers.
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So that means we're going to have to get
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some real number out that we then like to
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turn around and put into the function g.
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Well, g is a quadratic function, and the
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domain of a quadratic function is all real numbers.
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So, we're not going to have any restrictions whatsoever.
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The domain of the composition of these two
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functions will be all real numbers.
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So, one thing that you should be wondering
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is: does it matter which order you do the f
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and the g in a composition? So, let's find out.
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Here we want to find the composition of f with g.
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By definition this is f of g of x.
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We know from above that g of x is 4 x
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squared plus 6, so we make that substitution.
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And we see that what we want now is to find
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f of 4 x squared plus 6.
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So, look at f of x, and everywhere you see
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an x, replace it with 4 x squared plus 6.
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We get 3 times 4 x squared plus 6. Distribute.
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And we're also again asked to find the
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domain, but we're dealing with the same two
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functions, and we're going to take an x
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value starting with g of x.
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So, the domain of g of x is all real numbers.
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We're going to take whatever we get it and
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put it into the function f, a linear
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function whose domain is all real numbers,
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so again there aren't going to be any
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restrictions in what x can be, and the
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domain of this function is all real numbers again.
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Remember I asked you a minute ago if it
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mattered if you switch the order of the f
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and the g, and you can clearly see that you
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get a different answer in most cases.
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Sometimes you won't, and that's a very
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special type of function that will come up
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later in your studies.