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### Course: Precalculus > Unit 1

Lesson 1: Composing functions- 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)

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# Composing 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 $f(x)=3x-1$ and $g(x)={x}^{3}+2$ , then what is $f(g(3))$ ?

### Solution

One way to evaluate $f(g(3))$ is to work from the "inside out". In other words, let's evaluate $g(3)$ first and then substitute that result into $f$ to find our answer.

Let's evaluate $g(3)$ .

Since $g(3)=29$ , then $f(g(3))=f(29)$ .

Now let's evaluate $f(29)$ .

It follows that $f(g(3))=f(29)=86$ .

# Finding the composite function

In the above example, function $g$ took $3$ to $29$ , and then function $f$ took $29$ to $86$ . Let's find the function that takes $3$ directly to $86$ .

To do this, we must $f(g(x))$ .

**compose**the two functions and find### Example

What is $f(g(x))$ ?

*For reference, remember that*$f(x)=3x-1$
and $g(x)={x}^{3}+2$ .### Solution

If we look at the expression$f({g(x)})$ , we can see that ${g(x)}$ is the input of function $f$ . So, let's substitute ${g(x)}$ everywhere we see ${x}$ in function $f$ .

Since $g(x)={x}^{3}+2$ , we can substitute ${x}^{3}+2$ in for $g(x)$ .

This new function should take $3$ directly to $86$ . 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 $f$ composed with function $g$ , we can write $f\circ g$ , read as "$f$ composed with $g$ ". This composition is defined by the following rule:

The diagram below shows the relationship between $(f\circ g)(x)$ and $f(g(x))$ .

Now let's look at another example with this new definition in mind.

### Example

Find $(h\circ g)(x)$ and $(h\circ g)(-2)$ .

### Solution

We can find $(h\circ g)(x)$ as follows:

Since we now have function $h\circ g$ , we can simply substitute $-2$ in for $x$ to find $(h\circ g)(-2)$ .

Of course, we could have also found $(h\circ g)(-2)$ by evaluating $h(g(-2))$ . This is shown below:

The diagram below shows how $(h\circ g)(-2)$ is related to $h(g(-2))$ .

Here we can see that function $g$ takes $-2$ to $2$ and then function $h$ takes $2$ to $0$ , while function $h\circ g$ takes $-2$ directly to $0$ .

# Now let's practice some problems

### Problem 3

In problems 4 and 5, let $f(t)=t-2$ and $g(t)={t}^{2}+5$ .

### 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.(131 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)).(18 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)