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Course: College Algebra > Unit 11
Lesson 3: Composing functionsComposing functions
Walk through examples, explanations, and practice problems to learn how to find and evaluate composite functions.
Given two functions, we can combine them in such a way so that the outputs of one function become the inputs of the other. This action defines a composite function. Let's take a look at what this means!
Evaluating composite functions
Example
If and , then what is ?
Solution
One way to evaluate is to work from the "inside out". In other words, let's evaluate first and then substitute that result into to find our answer.
Let's evaluate .
Since , then .
Now let's evaluate .
It follows that .
Finding the composite function
In the above example, function took to , and then function took to . Let's find the function that takes directly to .
To do this, we must compose the two functions and find .
Example
What is ?
For reference, remember that
and .
For reference, remember that
Solution
If we look at the expression , we can see that is the input of function . So, let's substitute everywhere we see in function .
Since , we can substitute in for .
This new function should take directly to . Let's verify this.
Excellent!
Let's practice
Problem 1
Problem 2
Composite functions: a formal definition
In the above example, we found and evaluated a composite function.
In general, to indicate function composed with function , we can write , read as " composed with ". This composition is defined by the following rule:
The diagram below shows the relationship between and .
Now let's look at another example with this new definition in mind.
Example
Find and .
Solution
We can find as follows:
Since we now have function , we can simply substitute in for to find .
Of course, we could have also found by evaluating . This is shown below:
The diagram below shows how is related to .
Here we can see that function takes to and then function takes to , while function takes directly to .
Now let's practice some problems
Problem 3
In problems 4 and 5, let and .
Problem 4
Problem 5
Challenge Problem
Want to join the conversation?
- In practice Q 4, where is 4t created? I see where t^2 and 4 come from, but am not sure what puts 4t in(69 votes)
- I was stuck on this too, but I think the reason is that (t-2)^2 = (t-2)(t-2) . Using the distributive property, you get t^2-4t+4.(122 votes)
- (f ∘ g)(x)
here, what does the sign ∘ mean?(2 votes)- (f ∘ g)(x) is read "f of g of x", so the ∘ translates to "of".
In this case, if you had functions defined, f(x) and g(x), then to get (f ∘ g)(x) you would substitute g(x) for x inside of f(x). Another way to write it is f(g(x)).(15 votes)
- How do you know when to use the "inside out property" or the composing function?(9 votes)
- It doesn't really matter --- they will both give the same answer, so it's up to you to choose what works best/easiest for you with the problem you're given at the time!
(But, of course, you need to be familiar with both techniques.)(7 votes)
- May someone please explain the challenge problem to me?(4 votes)
- The challenge problem says, "The graphs of the equations y=f(x) and y=g(x) are shown in the grid below." So basically the two graphs is a visual representation of what the two different functions would look like if graphed and they're asking us to find (f∘g)(8), which is combining the two functions and inputting 8. From the definition, we know (f∘g)(8)=f(g(8)). So let's work "inside out". If we look at the graph of "g", we see that g(8) is 2 (look at the 8 at the x-axis and if you go up to where it meets the line, the y value would be 2). Because g(8)=2, then when you substitute it back in the equation, f(g(8)) would equal f(2). Then if we look at the graph of "f", we can see that f(2) is -3. (when you look at the 2 in the x-axis, it will correspond to -3 on the y-axis). So by looking at the graph, you can figure out that (f∘g)(8) is approximately -3.
~Dylan(15 votes)
- In question 4 how do people get the 4t in tsquered-t4+9?(3 votes)
- It comes from (t-2)^2
(t-2)^2 = (t-2)(t-2) = t^2-2t-2t+4 = t^2-4t+4
To square binomials, you need to use FOIL or the pattern for creating a perfect square trinomial. You can't square the 2 terms and get the right answer.
Hope this helps.(11 votes)
- in the example question "g(x)= x+4, h(x)= x(squared)-2x" how does it get the +8x and -2x in the distribute section ?
here's the distribute equation =(x(squared)+8x+16−2x−8)
(5 votes)- h(g(x)) = (x+4)^2 - 2(x+4)
Basically each "x" in h(x) gets replaced with (x+4), which if g(x). Then, you simplify.
1) FOIL out (x+4)^2:
h(g(x)) = x^2+4x+4x+16 - 2(x+4) = x^2 + 8x + 16 - 2(x+4)
2) Distribute -2: h(g(x)) = x^2 + 8x + 16 - 2x - 8
3) Combine like terms: x^2 + 6x + 8
Hope this helps.(6 votes)
- I still can't get this. I think my problem is them showing multiple ways to do this instead of focusing on how to combine it into one equation. Either I have to work each function alone then combine them at the end or have more help figuring out how to make one equation.(2 votes)
- I don't think their aim is to show you the multiple ways you can evaluate the composite function.
The first example they basically show what evaluating a composite function really means, it's like you said "work each function alone". In the second example they showed a more faster and efficient way to evaluate the composite function by combining them into one equation.
If you're still confused about composite functions, I'll explain this way:
we have a function f(x), this function takes "x" as "input". Now, I'm certain you're used to the variable x being substituted for a number, but in maths, you can pretty much substitute it for anything you like. (Expressions for example)
Like I can let x = 5, but I can also let x = 2h. Doesn't that mean I can also substitute x for some function? In other words x = g(x).
Say if g(k) = 4k, then this would become: x = 4k. (Because x = g(k) = 4k)
Since we let x = g(k) = 4k, then our function f can be written as: f( g(k) ) or f(5k) (We substituted x for g(k) )
if f(x) = 5x, by substituting x for g(k), this becomes:
f( g(x) ) = 5g(x) ---> f( 4k ) = 5(4k) = 20k
This also means that our composite function changes value depending on the value of k.
Conclusion: g(k) becomes input for function f.(8 votes)
- Can someone please simplify all of this for me cause i am so confused!(2 votes)
- Sometimes it's useful to look at a different point of view. Try this site. Then come back and try this video again. http://www.mathsisfun.com/sets/functions-composition.html(6 votes)
- Number 3 is hard can u give better explanations(4 votes)
- The way I understand it and I solve it is to always split solution in to steps where each step is solving just single function:
f(x) = 3x-5
g(x) = 3-2x
(g∘f)(3)
1. We'll solve f(x) as it's on the end. We know that x is 3 so we need to calculate 3*3-5 which is 4
2. We'll solve g(x). g(x) is wrapping up f(x) so it might look something like g(f(x)) = 3-2(fx) = 3-2(3x-5).
As we know from step 1 that f(x) = 4 we can just use it as x variable for g. So equation should be g(x) = 3-2*4
Esentially you can just focus on single function and use your result as x of next function.
I hope this is helpful and not more confusing.(2 votes)
- If f(x)=(1/x) and (f/g)(x)=((x+4)/(x^2+2x)), what is the function g?(4 votes)
- Based upon the rules for dividing with fractions: f/g = (1/x) / g = (1/x) * the reciprocal of g
We need to work in reverse
1) Factor denominator to undo the multiplication:(x+4)/(x^2+2x)
=(x+4)/[x(x+2)]
We can see there is a factor of X in the denominator. This would have been from multiplying 1/x * the reciprocal of g.
2) Separate the factor 1/x:(1/x) * (x+4)/(x+2)
This tells us the reciprocal of g =(x+4)/(x+2)
3) Flip it to find g:g(x) = (x+2)/(x+4)
Hope this helps.(2 votes)