# 1.4 Composition of functions  (Page 5/9)

 Page 5 / 9

Using [link] , evaluate $\text{\hspace{0.17em}}g\left(f\left(2\right)\right).$

$g\left(f\left(2\right)\right)=g\left(5\right)=3$

## Evaluating composite functions using formulas

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.

While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition $\text{\hspace{0.17em}}f\left(g\left(x\right)\right).\text{\hspace{0.17em}}$ To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like $\text{\hspace{0.17em}}f\left(t\right)={t}^{2}-t,\text{\hspace{0.17em}}$ we substitute the value inside the parentheses into the formula wherever we see the input variable.

Given a formula for a composite function, evaluate the function.

1. Evaluate the inside function using the input value or variable provided.
2. Use the resulting output as the input to the outside function.

## Evaluating a composition of functions expressed as formulas with a numerical input

Given $\text{\hspace{0.17em}}f\left(t\right)={t}^{2}-t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)=3x+2,\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}f\left(h\left(1\right)\right).$

Because the inside expression is $\text{\hspace{0.17em}}h\left(1\right),\text{\hspace{0.17em}}$ we start by evaluating $\text{\hspace{0.17em}}h\left(x\right)\text{\hspace{0.17em}}$ at 1.

$\begin{array}{l}h\left(1\right)=3\left(1\right)+2\\ h\left(1\right)=5\end{array}$

Then $\text{\hspace{0.17em}}f\left(h\left(1\right)\right)=f\left(5\right),\text{\hspace{0.17em}}$ so we evaluate $\text{\hspace{0.17em}}f\left(t\right)\text{\hspace{0.17em}}$ at an input of 5.

$\begin{array}{l}f\left(h\left(1\right)\right)=f\left(5\right)\\ f\left(h\left(1\right)\right)={5}^{2}-5\\ f\left(h\left(1\right)\right)=20\end{array}$

Given $\text{\hspace{0.17em}}f\left(t\right)={t}^{2}-t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)=3x+2,\text{\hspace{0.17em}}$ evaluate

1. $h\left(f\left(2\right)\right)$
2. $h\left(f\left(-2\right)\right)$

a. 8; b. 20

## Finding the domain of a composite function

As we discussed previously, the domain of a composite function such as $\text{\hspace{0.17em}}f\circ g\text{\hspace{0.17em}}$ is dependent on the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as $\text{\hspace{0.17em}}f\circ g.\text{\hspace{0.17em}}$ Let us assume we know the domains of the functions $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ separately. If we write the composite function for an input $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}f\left(g\left(x\right)\right),\text{\hspace{0.17em}}$ we can see right away that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ must be a member of the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ must be a member of the domain of $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ otherwise the second function evaluation in $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)\text{\hspace{0.17em}}$ cannot be completed, and the expression is still undefined. Thus the domain of $\text{\hspace{0.17em}}f\circ g\text{\hspace{0.17em}}$ consists of only those inputs in the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ that produce outputs from $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ belonging to the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ Note that the domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ composed with $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is the set of all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is in the domain of $\text{\hspace{0.17em}}f.$

## Domain of a composite function

The domain of a composite function $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)\text{\hspace{0.17em}}$ is the set of those inputs $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is in the domain of $\text{\hspace{0.17em}}f.$

Given a function composition $\text{\hspace{0.17em}}f\left(g\left(x\right)\right),$ determine its domain.

1. Find the domain of $\text{\hspace{0.17em}}g.$
2. Find the domain of $\text{\hspace{0.17em}}f.$
3. Find those inputs $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ That is, exclude those inputs $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ from the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is not in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ The resulting set is the domain of $\text{\hspace{0.17em}}f\circ g.$

## Finding the domain of a composite function

Find the domain of

The domain of $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ consists of all real numbers except $\text{\hspace{0.17em}}x=\frac{2}{3},\text{\hspace{0.17em}}$ since that input value would cause us to divide by 0. Likewise, the domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ consists of all real numbers except 1. So we need to exclude from the domain of $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ that value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}g\left(x\right)=1.$

$\begin{array}{l}\frac{4}{3x-2}=1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4=3x-2\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6=3x\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=2\hfill \end{array}$

So the domain of $\text{\hspace{0.17em}}f\circ g\text{\hspace{0.17em}}$ is the set of all real numbers except $\text{\hspace{0.17em}}\frac{2}{3}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ This means that

$x\ne \frac{2}{3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ne 2$

We can write this in interval notation as

$\left(-\infty ,\frac{2}{3}\right)\cup \left(\frac{2}{3},2\right)\cup \left(2,\infty \right)$

#### Questions & Answers

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert