Algebra (all content)
- Intro to composing functions
- Intro to composing functions
- Composing functions
- Evaluating composite functions
- Evaluate composite functions
- Evaluating composite functions: using tables
- Evaluating composite functions: using graphs
- Evaluate composite functions: graphs & tables
- Finding composite functions
- Find composite functions
- Evaluating composite functions (advanced)
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
If and , then what is ?
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 .
What is ?
For reference, remember that and .
For reference, remember that and .
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.
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.
Find and .
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
In problems 4 and 5, let and .
The graphs of the equations and are shown in the grid below.
Which of the following best approximates the value of ?
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(55 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.(93 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)).(10 votes)
- How do you know when to use the "inside out property" or the composing function?(8 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.)(6 votes)
- May someone please explain the challenge problem to me?(2 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.
- 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)
- 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:
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 =
3) Flip it to find g:
g(x) = (x+2)/(x+4)
Hope this helps.(2 votes)
- In problem 4, i got t^2 +9, so where did the -4t come from? I tried to find out back in the examples but it did the same. Its like taking the number inside the parenthesis and multiplying it by 2t or 2 times the variable.
=(x+4)^2−2(x+4) =x^2+8x+16−2x−8...this was the example, so where did 8x come from?(3 votes)
- Your mistake was simple. You didn't go about the exponent property the correct way. This is how I solved it:
at this point (t-2)^2, is where you made the error. I'm sure you probably distributed the 2 instead of multiplying (t-2) twice. But instead, FOIL it, and you get the quadratic function: t^2-4t+4.
Hence, where "-4t" came from. (:(2 votes)
- In the last question, is there a problem with the graph because, I'm finding it difficult to understand it?(2 votes)
- The graph is fine, but this question pools together some concepts from other parts of math the we may have forgotten about.
f(x)is your standard linear equation of the form
f(x) = mx + b
f(x) = -0.5x - 2
g(x) is a sinusoidal equation of the form
g(x) = a cos(b (x - c)) + d
ais the amplitude,
2π/|b|is the period,
cis the phase shift, and
dis the midline.
ais simple enough; it's
2. The phase shift and midline both look like
0, so that's nice. The tricky part is finding
The period of this particular sinusoidal function appears to be
8, which means we need to solve for
2π/|b| = 8
# Do some algebra...
b = π/4
Once we have done this we can compose the functions like so
f(x) = -0.5 * x - 2
g(x) = 2 * cos(π / 4 * x)
f(g(x)) = -0.5 * (2 * cos(π / 4 * x)) - 2
Which we can further simplify as
f(g(x)) = -cos(π / 4 * x) - 2
Of course you could forgo composing the functions and evaluate each function separately. Either way, you'll find that
f(g(x)) ~= -3.
Here's a graph of everything: https://www.desmos.com/calculator/b0r64jqiqv(4 votes)
- For the challenge problem, since g(8)=2 why does f(g(8)) also equal f(2)? That part just really confused me(1 vote)
- Because g(8) = 2, we can substitute 2 for g(8) anywhere it appears.
That's what has happened in f(g(8)) to get f(2).(4 votes)
- what if its f o f (x)? will it be f(x)^2?
i heard that there's a definition that states that
f o f o f ....(n times)= f(x)^n
is it true?(2 votes)
- In some contexts, composing f n times is written as fⁿ(x), but we don't usually use this notation in K-12 or on Khan Academy. f²(x) usually means the product f(x)·f(x). sin²(x) is used this way.(2 votes)