Question 1

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

Accepted Answer

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.

Question 2

How do you know when to use the "inside out property" or the composing function?

Accepted Answer

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.)

Question 3

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)
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Accepted Answer

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.

Question 4

If f(x)=(1/x) and (f/g)(x)=((x+4)/(x^2+2x)), what is the function g?

Accepted Answer

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.

Question 5

In question 4 how do people get the 4t in tsquered-t4+9?

Accepted Answer

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.

Question 6

How do we know that g = 3 in the first example study? I looked multiple times, and couldn't see where we found that value. Any help?

Accepted Answer

I assume you are asking about the first example on the page.  The initial problem statement gives you the equations for f(x) and g(x).  It then asks you to find f(g(3)).

g(3) is part of what the problem is asking you to find.  It doesn't say that g=3.  It says uses the function g(x) with an input value of x=3.

Hope this clarifies thing.

Question 7

(f ∘ g)(x)
here, what does the sign ∘ mean?

Accepted Answer

(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)).

Question 8

May someone please explain the challenge problem to me?

Accepted Answer

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

Question 9

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.

Accepted Answer

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.

Question 10

Can someone please simplify all of this for me cause i am so confused!

Accepted Answer

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

Course: Precalculus > Unit 1

Composing functions

Evaluating composite functions

Example

Solution

Finding the composite function

Example

Solution

Let's practice

Problem 1

Problem 2

Composite functions: a formal definition

Example

Solution

Now let's practice some problems

Problem 3

Problem 4

Problem 5

Challenge Problem

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