# 1.1 Functions and function notation  (Page 10/21)

 Page 10 / 21

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

## Algebraic

For the following exercises, determine whether the relation represents a function.

$\left\{\left(a,b\right),\left(b,c\right),\left(c,c\right)\right\}$

function

For the following exercises, determine whether the relation represents $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$

$5x+2y=10$

$y={x}^{2}$

function

$x={y}^{2}$

$3{x}^{2}+y=14$

function

$2x+{y}^{2}=6$

$y=-2{x}^{2}+40x$

function

$y=\frac{1}{x}$

$x=\frac{3y+5}{7y-1}$

function

$x=\sqrt{1-{y}^{2}}$

$y=\frac{3x+5}{7x-1}$

function

${x}^{2}+{y}^{2}=9$

$2xy=1$

function

$x={y}^{3}$

$y={x}^{3}$

function

$y=\sqrt{1-{x}^{2}}$

$x=±\sqrt{1-y}$

function

$y=±\sqrt{1-x}$

${y}^{2}={x}^{2}$

not a function

${y}^{3}={x}^{2}$

For the following exercises, evaluate the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ at the indicated values

$f\left(x\right)=2x-5$

$\begin{array}{cccc}f\left(-3\right)=-11;& f\left(2\right)=-1;& f\left(-a\right)=-2a-5;& -f\left(a\right)=-2a+5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(a+h\right)=2a+2h-5\end{array}$

$f\left(x\right)=-5{x}^{2}+2x-1$

$f\left(x\right)=\sqrt{2-x}+5$

$\begin{array}{cccc}f\left(-3\right)=\sqrt{5}+5;& f\left(2\right)=5;& f\left(-a\right)=\sqrt{2+a}+5;& -f\left(a\right)=-\sqrt{2-a}-5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(a+h\right)=\end{array}$ $\sqrt{2-a-h}+5$

$f\left(x\right)=\frac{6x-1}{5x+2}$

$f\left(x\right)=|x-1|-|x+1|$

Given the function $\text{\hspace{0.17em}}g\left(x\right)=5-{x}^{2},\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}\frac{g\left(x+h\right)-g\left(x\right)}{h},\text{\hspace{0.17em}}h\ne 0.$

Given the function $\text{\hspace{0.17em}}g\left(x\right)={x}^{2}+2x,\text{\hspace{0.17em}}$ evaluate $\text{\hspace{0.17em}}\frac{g\left(x\right)-g\left(a\right)}{x-a},\text{\hspace{0.17em}}x\ne a.$

$\frac{g\left(x\right)-g\left(a\right)}{x-a}=x+a+2,\text{\hspace{0.17em}}x\ne a$

Given the function $\text{\hspace{0.17em}}k\left(t\right)=2t-1:$

1. Evaluate $\text{\hspace{0.17em}}k\left(2\right).$
2. Solve $\text{\hspace{0.17em}}k\left(t\right)=7.$

Given the function $\text{\hspace{0.17em}}f\left(x\right)=8-3x:$

1. Evaluate $\text{\hspace{0.17em}}f\left(-2\right).$
2. Solve $\text{\hspace{0.17em}}f\left(x\right)=-1.$

a. $\text{\hspace{0.17em}}f\left(-2\right)=14;\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}x=3$

Given the function $\text{\hspace{0.17em}}p\left(c\right)={c}^{2}+c:$

1. Evaluate $\text{\hspace{0.17em}}p\left(-3\right).$
2. Solve $\text{\hspace{0.17em}}p\left(c\right)=2.$

Given the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}-3x:$

1. Evaluate $\text{\hspace{0.17em}}f\left(5\right).$
2. Solve $\text{\hspace{0.17em}}f\left(x\right)=4.$

a. $\text{\hspace{0.17em}}f\left(5\right)=10;\text{\hspace{0.17em}}$ b. or

Given the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x+2}:$

1. Evaluate $\text{\hspace{0.17em}}f\left(7\right).$
2. Solve $\text{\hspace{0.17em}}f\left(x\right)=4.$

Consider the relationship $\text{\hspace{0.17em}}3r+2t=18.$

1. Write the relationship as a function $\text{\hspace{0.17em}}r=f\left(t\right).$
2. Evaluate $\text{\hspace{0.17em}}f\left(-3\right).$
3. Solve $\text{\hspace{0.17em}}f\left(t\right)=2.$

a. $\text{\hspace{0.17em}}f\left(t\right)=6-\frac{2}{3}t;\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}f\left(-3\right)=8;\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}t=6\text{\hspace{0.17em}}$

## Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

not a function

function

function

function

function

function

Given the following graph,

• Evaluate $\text{\hspace{0.17em}}f\left(-1\right).$
• Solve for $\text{\hspace{0.17em}}f\left(x\right)=3.$

Given the following graph,

• Evaluate $\text{\hspace{0.17em}}f\left(0\right).$
• Solve for $\text{\hspace{0.17em}}f\left(x\right)=-3.$

a. $\text{\hspace{0.17em}}f\left(0\right)=1;\text{\hspace{0.17em}}$ b. or

Given the following graph,

• Evaluate $\text{\hspace{0.17em}}f\left(4\right).$
• Solve for $\text{\hspace{0.17em}}f\left(x\right)=1.$

For the following exercises, determine if the given graph is a one-to-one function.

not a function so it is also not a one-to-one function

one-to- one function

function, but not one-to-one

## Numeric

For the following exercises, determine whether the relation represents a function.

$\left\{\left(-1,-1\right),\left(-2,-2\right),\left(-3,-3\right)\right\}$

$\left\{\left(3,4\right),\left(4,5\right),\left(5,6\right)\right\}$

function

$\left\{\left(2,5\right),\left(7,11\right),\left(15,8\right),\left(7,9\right)\right\}$

For the following exercises, determine if the relation represented in table form represents $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.$

 $x$ 5 10 15 $y$ 3 8 14

function

 $x$ 5 10 15 $y$ 3 8 8
 $x$ 5 10 10 $y$ 3 8 14

not a function

For the following exercises, use the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ represented in [link] .

 $x$ $f\left(x\right)$ 0 74 1 28 2 1 3 53 4 56 5 3 6 36 7 45 8 14 9 47

Evaluate $\text{\hspace{0.17em}}f\left(3\right).$

Solve $\text{\hspace{0.17em}}f\left(x\right)=1.$

$f\left(x\right)=1,\text{\hspace{0.17em}}x=2$

For the following exercises, evaluate the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ at the values $f\left(-2\right),\text{\hspace{0.17em}}f\left(-1\right),\text{\hspace{0.17em}}f\left(0\right),\text{\hspace{0.17em}}f\left(1\right),$ and $\text{\hspace{0.17em}}f\left(2\right).$

$f\left(x\right)=4-2x$

$f\left(x\right)=8-3x$

$\begin{array}{ccccc}f\left(-2\right)=14;& f\left(-1\right)=11;& f\left(0\right)=8;& f\left(1\right)=5;& f\left(2\right)=2\end{array}$

$f\left(x\right)=8{x}^{2}-7x+3$

$f\left(x\right)=3+\sqrt{x+3}$

$\begin{array}{ccccc}f\left(-2\right)=4;\text{ }& f\left(-1\right)=4.414;& f\left(0\right)=4.732;& f\left(1\right)=4.5;& f\left(2\right)=5.236\end{array}$

$f\left(x\right)=\frac{x-2}{x+3}$

$f\left(x\right)={3}^{x}$

$\begin{array}{ccccc}f\left(-2\right)=\frac{1}{9};& f\left(-1\right)=\frac{1}{3};& f\left(0\right)=1;& f\left(1\right)=3;& f\left(2\right)=9\end{array}$

For the following exercises, evaluate the expressions, given functions $f,\text{\hspace{0.17em}}\text{\hspace{0.17em}}g,$ and $\text{\hspace{0.17em}}h:$

• $f\left(x\right)=3x-2$
• $g\left(x\right)=5-{x}^{2}$
• $h\left(x\right)=-2{x}^{2}+3x-1$

$3f\left(1\right)-4g\left(-2\right)$

$f\left(\frac{7}{3}\right)-h\left(-2\right)$

20

## Technology

For the following exercises, graph $\text{\hspace{0.17em}}y={x}^{2}\text{\hspace{0.17em}}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

$\left[-100,100\right]$

For the following exercises, graph $\text{\hspace{0.17em}}y={x}^{3}\text{\hspace{0.17em}}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

For the following exercises, graph $\text{\hspace{0.17em}}y=\sqrt{x}\text{\hspace{0.17em}}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

For the following exercises, graph $y=\sqrt[3]{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

$\left[-0.001,\text{0.001}\right]$

$\left[-0.1,\text{0.1}\right]$

$\left[-1000,\text{1000}\right]$

$\left[-1,000,000,\text{1,000,000}\right]$

## Real-world applications

The amount of garbage, $\text{\hspace{0.17em}}G,\text{\hspace{0.17em}}$ produced by a city with population $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is given by $\text{\hspace{0.17em}}G=f\left(p\right).\text{\hspace{0.17em}}$ $G\text{\hspace{0.17em}}$ is measured in tons per week, and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is measured in thousands of people.

1. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$
2. Explain the meaning of the statement $\text{\hspace{0.17em}}f\left(5\right)=2.$

The number of cubic yards of dirt, $\text{\hspace{0.17em}}D,\text{\hspace{0.17em}}$ needed to cover a garden with area $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ square feet is given by $\text{\hspace{0.17em}}D=g\left(a\right).$

1. A garden with area 5000 ft 2 requires 50 yd 3 of dirt. Express this information in terms of the function $\text{\hspace{0.17em}}g.$
2. Explain the meaning of the statement $\text{\hspace{0.17em}}g\left(100\right)=1.$

a. $\text{\hspace{0.17em}}g\left(5000\right)=50;$ b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.

Let $\text{\hspace{0.17em}}f\left(t\right)\text{\hspace{0.17em}}$ be the number of ducks in a lake $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years after 1990. Explain the meaning of each statement:

1. $f\left(5\right)=30$
2. $f\left(10\right)=40$

Let $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ be the height above ground, in feet, of a rocket $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds after launching. Explain the meaning of each statement:

1. $h\left(1\right)=200$
2. $h\left(2\right)=350$

a. The height of a rocket above ground after 1 second is 200 ft. b. the height of a rocket above ground after 2 seconds is 350 ft.

Show that the function $\text{\hspace{0.17em}}f\left(x\right)=3{\left(x-5\right)}^{2}+7\text{\hspace{0.17em}}$ is not one-to-one.

find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
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
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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
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