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Evaluating composite functions: using tables

Given tables of values of the functions f and g, Sal evaluates f(g(0)) and g(f(0)).

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• I really, really don't get this. And I can't graduate unless I get through Algebra 2. How does the graph stuff make up an equation?
• Sal is showing that f(x) and g(x) represents equations. We don't know what those equations are, instead we are only given their inputs and outputs.
So, for f(x) when x= -1 the output is 5. In other words: f(-1)=5

In the question above f(g(0)) means we are given 0 as an input for g(x), the output is 5, we then have f(5), the output of g(0) becomes the input for f(x), then the output of f(5)=11

Does that make any more sense?
• Hi, what if you are asked for something like this:

Two functions f and g are such that f(x)=1+2x and g(f(x))=x+3. What is the value of g(5)?

• Your function g(x) is defined as a combined function of g(f(x)), so you don't have a plain g(x) that you can just evaluate using 5. The 5 needs to be the output from f(x). So, start by finding: 5=1+2x
That get's you back to the original input value that you can then use as the input to g(f(x)).

Subtract 1: 4=2x
Divided by 2: x=2

Now, use 2 as the input to g(f(x))=2+3 = 5

I think this is right. Maybe someone else can verify it.
• So I hope someone can understand this without actually seeing the tables I'm working with but here's a problem I'm stumped on:
"make a table of values for f(x) after the given translation: 3 units up."
It then gives me a table with a column "x" and a column "f(x)" with functions in both. then there is an empty column I'm supposed to fill in with the name "g(x)". So, what is the process to complete the function with the criteria of "3 units up"? Can one of you that are actually smart help me out please?
• I'm not certain but I think "3 units up" could mean either of these 2 things :-
1. The graph of f(x) has been vertically moved up in the y axis by 3. You have to find the new line & from there get the outputs for the empty column.
2. It could be asking for f(x+3) in which case find the equation for f(x) & substitute all the x in the equation with x+3.
• Hi there, can someone tell me what level of math this is (e.g. what high school grade level (grade 10/11/12? or AP)? Many thanks!
• It depends on the country. If you are asking about the US, then it depends on the state. I learned this in 7th grade when I took Algebra I, but it is possible to not reach it until maybe 10th or even 11th grade depending on several factors.
• I want to know how to solve f(x) and g(x) using a table without a value, like in this video it shows 0, i want to solve without any value.
• For that you need to know f(x) and g(x) in terms of x(like f(x)=3x+ln(x) for example) so that you can evaluate them at any value.
In the type of questions where f(x) and g(x) are not given in terms of x, the table of values must be used to evaluate them and their composite functions. This can only be done at the x values given in the table.
(1 vote)
• okay so im working on this math AIR packet and one question is

Ryan works for a delivery service. The function f(n) is used to calculate his daily pay, in dollars, on a day when he makes (n) deliveries
f(n)=7n+96
and the table gives me the first two numbers of deliveries which is 0 and
5
and I figured out the daily pay for both, for f(0)=96 and f(5)=277 and it only gives me the daily pay after that and I don't know how to figure out the number of deliveries
(1 vote)
• Please tell me: do you just take a number on the graph when the problem asks for it and plug in?
(1 vote)
• at how would be f of 4 be 27
(1 vote)
• There is no f of 4 in this problem, only f of -4. It shows that f of -4 is 29, not 27.
(1 vote)
• How do you find the range for the combined function?
(1 vote)
• Is this rule same for all table functions cause I am a new learner so I need to know a lot about functions?
(1 vote)

Video transcript

- [Voiceover] So we have some tables here that give us what the functions f and g are when you give it certain inputs. So, when you input negative four, f of negative four is 29. That's going to be the output of that function. So we have that for both f and g, and what I want to do is evaluate two composite functions. I want to evaluate f of g of zero, and I want to evaluate g of f of zero. So like always, pause the video and see if you can figure it out. Let's first think about f of g of zero. F of g of zero. What is this all about? Actually let me use multiple colors here. F of g of zero. Well, this means that we're going to evaluate g at zero, so we're gonna input zero into g. Do it in that. So we're gonna input zero into our function g, and we're going to output, whatever we output is going to be g of zero. I'll write it right over here, and then we're going to input that into our function f. We're going to input that into our function f, and whatever I output then is going to be f of g of zero. F of g of zero. F of g of zero. I wrote these small here so we have space for the actual values. So first let's just evaluate, and if you are now inspired, pause the video again and see if you can solve it. Although, if you solved it the first time, you don't have to do that now. What's g of zero? Well, when we input x equals zero, we get g of zero is equal to five. So g of zero is five. So that is five. So we're now going to input five into our function f. We're essentially going to evaluate f of five. So when you input five into our function. I'm gonna do it in this brown color. When you input x equals five into f, you get the function f of five is equal to 11. So this is going to be 11. So, f of g of zero is equal to 11. Now, let's do g of f of zero. So now let's evaluate. I'll do this is different colors. G, maybe I'll use those same two colors actually. So now we're going to evaluate g of f of zero. G of f of zero, and the key realization is you wanna go within the parenthesis. Evaluate that first so then you can evaluate the function that's kind of on the outside. So here we're going to take zero as an input into the function f, and then whatever that is, that f of zero, we're going to input into our function g. We're going to input into our function g, and what we're going to be, and then the output of that is going to be g of f of zero. So, let's see, what is f of zero? You see over here when our input is zero, this table tells us that f of zero is equal to one. So f of zero is equal to one. F of zero is equal to one. So now we use one as an input into g. We're now evaluating g of one, or I can just write this. This is the same thing as g of one. G of one. Once again, why was that? 'Cause f of zero is equal to, f of zero is equal to one. And let me, I wrote those parenthesis too far away from the g. This is the same thing as g of one. Because once again f of zero is one. Now what is g of one? Well, when I input one into our function g, I get g of one is equal to eight. So this is going to be equal, this is equal to eight, and we're done. And notice these are different values, because these are different composite functions. F of g of zero is 11, and g of f of zero is eight.