For Part 2 evaluating a combined function, method 1 where did the 2 come from

We are substituting x=2 into the function.

Are these functions like functions in programing in which you can write the function beforehand and then call it whenever you want?

Basically yes ,they are. You can create them based on the information given to you and then you can add, subtract, multiply, and divide it. But you don't neccesarily "call" the function rather you input a given value into the function. So they are similar to each other, but not the exact same.

on part 2 evaluating a combined function do you always replace x wit 2 or not becuse i am confused on how they got 2

Unless the function has a restricted domain, you can evaluate the function (including the combined function) for any value of "x". So, you will not always replace x with 2. You can evaluate the new combined function h(x) for any value of x. Sal just happened to use x=2 to demonstrate the process.

where does (-5) come in the last practice problem?

Hello Gabriela. Sorry for a late response. You probably understand the question by now, but I'm going to answer for you anyways. Basically, the (t) is like an x-value. If a number replaces the (t) then that is your x-value or your input. The letter on the left of the r(t) is your y-value. The 'r' in r(t) =..... is your y-value/output. Your 't' is your input/x-value. The 'p' in p(t) =..... is your y-value/output. Your 't' is your input/x-value. The 'q' in q(t) =..... is your y-value/output. Your 't' is your input/x-value. Of course just remember the positions and all because the letters and values changes. Lol. In this case p(3) = t + 2 is synonymous for p = 3 + 2. Remember p is synonymous for y. So in this equation it is p = 5. In q(3) = t - 1 Remember q is synonymous for y. It is same thing for THIS scenario. Plug in 3 into your x-value or 't'. They are the same thing. So it is q = 3 - 1. q =2 In r(3) = t. Essentially whatever is r(t) = t. So whatever is t on the left is t on the right. Think of it like (t) =t = r . (x) = x = y. So all the values are 3. Now you just plug your answers in the coordinating spots. (*Note the last 5 in the equation below is POSITIVE. So is the 3 before. You are just subtracting the two values.) 5 * 2 * 3 - 5 10*3 -5......Multiply first 5*2 30 - 5 ......Again, multiply 10*3 25.....Then subtract 30 -5 Your answer is 25. Hopefully you and others will understand.

How would I solve the equations Given f(x)=2x-3 and g(x)=0.5x+4 find f of g of x? [f of g](x)

In order to find [f of g](x), you need to substitute function g(x) into function f(x) (replace the x in f(x) with function g(x)). [f of g](x) = 2(0.5x+4)-3 = x+8-3 = x+5 Therefore, [f of g](x) = x+5

Will this be on the SAT for March 9th, 2019?

Function combination may be on the SAT. It depends on the specific SAT given, and we have no way of finding out if it's on the test, but you should prepare for it nonetheless.

for graphing f(3) + g(3) Why does f(3)=6 and g(3)=3? I see no indicator as to why f(3) should = 6

Graphically, for any function f(x), the statement that f(a)=b means that the graph of f(x) passes through the point (a,b). If you look at the graphs of f(x) and g(x), you will see that the graph of f(x) passes through the point (3,6) and the graph of g(x) passes though the point (3,3). This is why f(3)=6 and g(3)=3.

Im in Pre Cal right now...just looking back on Algebra II for the SAT...I got how the first 2 graphs worked...but to then add a parabola in the third question...just confuses me

Same here, I'm looking back just to refresh my memory. It's simple, just find the x-coordinate that it asks you to, in this case a 3 Then find where the line intercepts to find the y value, which is for the parabola part, a 6, use the y-coordinates and, in this case, add them together, poof, tell me what you get

Main content

Course: Algebra II (2018 edition) > Unit 1

Lesson 1: Combining functions

Intro to combining functions

Google Classroom

Become familiar with the idea that we can add, subtract, multiply, or divide two functions together to make a new function.

Just like we can add, subtract, multiply, and divide numbers, we can also add, subtract, multiply, and divide functions.

The sum of two functions

Part 1: Creating a new function by adding two functions

Let's add

f (x) = x + 1

and

g (x) = 2 x

together to make a new function.

$\begin{aligned} f (x) + g (x) & = (x + 1) + (2 x) \\ = x + 1 + 2 x \\ = 3 x + 1 \end{aligned}$ ‍

Let's call this new function

h

. So we have:

$h (x) = f (x) + g (x) = 3 x + 1$ ‍

Part 2: Evaluating a combined function

We can also evaluate combined functions for particular inputs. Let's evaluate function

h

above for

x = 2

. Below are two ways of doing this.

Method 1: Substitute

x = 2

into the combined function

h

$\begin{aligned} h (x) & = 3 x + 1 \\ h (2) & = 3 (2) + 1 \\ = 7 \end{aligned}$ ‍

Method 2: Find

f (2)

and

g (2)

and add the results.

Since

h (x) = f (x) + g (x)

, we can also find

h (2)

by finding

f (2) + g (2)

First, let's find

f (2)

$\begin{aligned} f (x) & = x + 1 \\ f (2) & = 2 + 1 \\ = 3 \end{aligned}$ ‍

Now, let's find

g (2)

$\begin{aligned} g (x) & = 2 x \\ g (2) & = 2 \cdot 2 \\ = 4 \end{aligned}$ ‍

f (2) + g (2) = 3 + 4 = 7

Notice that substituting

x = 2

directly into function

h

and finding

f (2) + g (2)

gave us the same answer!

Now let's try some practice problems.

In problems 1 and 2, let

f (x) = 3 x + 2

and

g (x) = x - 3

Problem 1

Find $f (x) + g (x)$ ‍.

Problem 2

Evaluate $f (- 1) + g (- 1)$ ‍.

Here are a couple of ways to find the answer.

Method 1: Find

f (- 1)

and

g (- 1)

separately. Then add the two.

First, let's find

f (- 1)

$\begin{aligned} f (x) & = 3 x + 2 \\ f (- 1) & = 3 (- 1) + 2 \\ = - 1 \end{aligned}$ ‍

Next, let's find

g (- 1)

$\begin{aligned} g (x) & = x - 3 \\ g (- 1) & = - 1 - 3 \\ = - 4 \end{aligned}$ ‍

f (- 1) + g (- 1) = - 1 + (- 4) = - 5

Method 2: Find

f (x) + g (x)

first, and then evaluate this expression when

x = - 1

$\begin{aligned} f (x) + g (x) & = (3 x + 2) + (x - 3) \\ = 3 x + 2 + x - 3 \\ = 4 x - 1 \end{aligned}$ ‍

When

x = - 1

, this expression is equal to

4 (- 1) - 1

- 5

A graphical connection

We can also understand what it means to add two functions by looking at graphs of the functions.

The graphs of

y = m (x)

and

y = n (x)

are shown below. In the first graph, notice that

m (4) = 2

. In the second graph, notice that

n (4) = 5

Let

p (x) = m (x) + n (x)

. Now look at the graph of

y = p (x)

. Notice that

p (4) = 2 + 5 = 7

Challenge yourself to see that

p (x) = m (x) + n (x)

for every value of

x

by looking at the three graphs.

Let's practice.

Problem 3

The graphs of

y = f (x)

and

y = g (x)

are shown below.

Which is the best approximation of $f (3) + g (3)$ ‍?

Other ways to combine functions

All of the examples we've looked at so far create a new function by adding two functions, but you can also subtract, multiply, and divide two functions to make new functions!

For example, if

f (x) = x + 3

and

g (x) = x - 2

, then we can not only find the sum, but also ...

... the difference.

\begin{aligned} f (x) - g (x) & = (x + 3) - (x - 2) Substitute. \\ = x + 3 - x + 2 Distribute negative sign. \\ = 5 Combine like terms. \end{aligned}

... the product.

\begin{aligned} f (x) \cdot g (x) & = (x + 3) (x - 2) Substitute. \\ = x^{2} - 2 x + 3 x - 6 Distribute. \\ = x^{2} + x - 6 Combine like terms. \end{aligned}

... the quotient.

\begin{aligned} f (x) \div g (x) & = \frac{f (x)}{g (x)} \\ = \frac{(x + 3)}{(x - 2)} Substitute. \end{aligned}

In doing so, we have just created three new functions!

Challenge problem

p (t) = t + 2

q (t) = t - 1

r (t) = t

Evaluate $p (3) \cdot q (3) \cdot r (3) - p (3)$ ‍.

Let's first find

p (3)

q (3)

, and

r (3)

by substituting

t = 3

into each of the formulas.

First, let's find

p (3)

$\begin{aligned} p (t) & = t + 2 \\ p (3) & = 3 + 2 \\ = 5 \end{aligned}$ ‍

Next, let's find

q (3)

$\begin{aligned} q (t) & = t - 1 \\ q (3) & = 3 - 1 \\ = 2 \end{aligned}$ ‍

Finally, let's find

r (3)

$\begin{aligned} r (t) & = t \\ r (3) & = 3 \\ = 3 \end{aligned}$ ‍

We can now find

p (3) \cdot q (3) \cdot r (3) - p (3)

as follows:

\begin{aligned} p (3) \cdot q (3) \cdot r (3) - p (3) & = 5 \cdot 2 \cdot 3 - 5 \\ = 30 - 5 \\ = 25 \end{aligned}

We could also find

p (t) \cdot q (t) \cdot r (t) - p (t)

first, and then find the value of this expression when

t = 3

. This would be more complicated, though, since the result would be a third degree polynomial!

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Junior Murray
Posted 6 years ago. Direct link to Junior Murray's post “For Part 2 evaluating a c...”
For Part 2 evaluating a combined function, method 1 where did the 2 come from
Button navigates to signup pageComment on Junior Murray's post “For Part 2 evaluating a c...”
(12 votes)
Answer
- Joshua Lee
  Posted 4 years ago. Direct link to Joshua Lee's post “We are substituting x=2 i...”
  We are substituting x=2 into the function.
  Button navigates to signup page
  (3 votes)
gabrielalopes2001.GL
Posted 6 years ago. Direct link to gabrielalopes2001.GL's post “where does (-5) come in t...”
where does (-5) come in the last practice problem?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Bailey Mitchell
  Posted 6 years ago. Direct link to Bailey Mitchell's post “Hello Gabriela. Sorry for...”
  Hello Gabriela. Sorry for a late response. You probably understand the question by now, but I'm going to answer for you anyways.
  
  Basically, the (t) is like an x-value. If a number replaces the (t) then that is your x-value or your input. The letter on the left of the r(t) is your y-value.
  The 'r' in r(t) =..... is your y-value/output. Your 't' is your input/x-value.
  The 'p' in p(t) =..... is your y-value/output. Your 't' is your input/x-value.
  The 'q' in q(t) =..... is your y-value/output. Your 't' is your input/x-value.
  
  Of course just remember the positions and all because the letters and values changes. Lol.
  
  In this case p(3) = t + 2 is synonymous for p = 3 + 2. Remember p is synonymous for y. So in this equation it is p = 5.
  
  In q(3) = t - 1 Remember q is synonymous for y. It is same thing for THIS scenario. Plug in 3 into your x-value or 't'. They are the same thing. So it is q = 3 - 1. q =2
  
  In r(3) = t. Essentially whatever is r(t) = t. So whatever is t on the left is t on the right. Think of it like (t) =t = r . (x) = x = y. So all the values are 3.
  
  Now you just plug your answers in the coordinating spots.
  (*Note the last 5 in the equation below is POSITIVE. So is the 3 before. You are just subtracting the two values.)
  5 * 2 * 3 - 5
  10*3 -5......Multiply first 5*2
  30 - 5 ......Again, multiply 10*3
  25.....Then subtract 30 -5
  Your answer is 25.
  
  Hopefully you and others will understand.
  Button navigates to signup page
  (15 votes)
Christopher M
Posted 6 years ago. Direct link to Christopher M's post “Are these functions like ...”
Are these functions like functions in programing in which you can write the function beforehand and then call it whenever you want?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Sujal J.👍
  Posted 6 years ago. Direct link to Sujal J.👍's post “Basically yes ,they are. ...”
  Basically yes ,they are. You can create them based on the information given to you and then you can add, subtract, multiply, and divide it. But you don't neccesarily "call" the function rather you input a given value into the function. So they are similar to each other, but not the exact same.
  Button navigates to signup page
  (5 votes)
Amira
Posted 6 years ago. Direct link to Amira's post “How would I solve the equ...”
How would I solve the equations
Given f(x)=2x-3 and g(x)=0.5x+4 find f of g of x? [f of g](x)
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Iris Nogueroles Langa
  Posted 6 years ago. Direct link to Iris Nogueroles Langa's post “In order to find [f of g]...”
  In order to find [f of g](x), you need to substitute function g(x) into function f(x) (replace the x in f(x) with function g(x)).
  
  [f of g](x) = 2(0.5x+4)-3 = x+8-3 = x+5
  
  Therefore, [f of g](x) = x+5
  Button navigates to signup page
  (8 votes)
dhebel612
Posted 7 years ago. Direct link to dhebel612's post “on part 2 evaluating a co...”
on part 2 evaluating a combined function do you always replace x wit 2 or not becuse i am confused on how they got 2
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “Unless the function has a...”
  Unless the function has a restricted domain, you can evaluate the function (including the combined function) for any value of "x". So, you will not always replace x with 2. You can evaluate the new combined function h(x) for any value of x. Sal just happened to use x=2 to demonstrate the process.
  Button navigates to signup page
  (5 votes)
desit241814
Posted 5 years ago. Direct link to desit241814's post “Will this be on the SAT f...”
Will this be on the SAT for March 9th, 2019?
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(2 votes)
Answer
- Alex
  Posted 5 years ago. Direct link to Alex's post “Function combination may ...”
  Function combination may be on the SAT. It depends on the specific SAT given, and we have no way of finding out if it's on the test, but you should prepare for it nonetheless.
  Button navigates to signup page
  (2 votes)
sf49
Posted 8 years ago. Direct link to sf49's post “I dont understand how the...”
I dont understand how the addition of the functions in problem 3 is 9 shouldnt it be 6?
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(1 vote)
Answer
KevinHJoe
Posted 5 years ago. Direct link to KevinHJoe's post “for graphing f(3) + g(3) ...”
for graphing f(3) + g(3)
Why does f(3)=6 and g(3)=3?

I see no indicator as to why f(3) should = 6
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Ian Pulizzotto
  Posted 5 years ago. Direct link to Ian Pulizzotto's post “Graphically, for any func...”
  Graphically, for any function f(x), the statement that f(a)=b means that the graph of f(x) passes through the point (a,b).
  
  If you look at the graphs of f(x) and g(x), you will see that the graph of f(x) passes through the point (3,6) and the graph of g(x) passes though the point (3,3). This is why f(3)=6 and g(3)=3.
  Comment on Ian Pulizzotto's post “Graphically, for any func...”
  (3 votes)
cpsnarayan
Posted 7 years ago. Direct link to cpsnarayan's post “I am a doing Advanced Alg...”
I am a doing Advanced Algebra 1 in sixth grade. Does that mean i do algebra 2 in eighth grade?
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(2 votes)
Answer
Lancelot.Quitian
Posted 7 years ago. Direct link to Lancelot.Quitian's post “Im in Pre Cal right now.....”
Im in Pre Cal right now...just looking back on Algebra II for the SAT...I got how the first 2 graphs worked...but to then add a parabola in the third question...just confuses me
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- V
  Posted 7 years ago. Direct link to V's post “Same here, I'm looking ba...”
  Same here, I'm looking back just to refresh my memory.
  It's simple, just find the x-coordinate that it asks you to, in this case a 3
  Then find where the line intercepts to find the y value, which is for the parabola part, a 6, use the y-coordinates and, in this case, add them together, poof, tell me what you get
  Comment on V's post “Same here, I'm looking ba...”
  (2 votes)