# 1.1 Functions and function notation  (Page 8/21)

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The most common graphs name the input value $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the output value $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ and we say $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is a function of $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}y=f\left(x\right)\text{\hspace{0.17em}}$ when the function is named $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ The graph of the function is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ in the plane that satisfies the equation $y=f\left(x\right).\text{\hspace{0.17em}}$ If the function is defined for only a few input values, then the graph of the function is only a few points, where the x -coordinate of each point is an input value and the y -coordinate of each point is the corresponding output value. For example, the black dots on the graph in [link] tell us that $\text{\hspace{0.17em}}f\left(0\right)=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(6\right)=1.\text{\hspace{0.17em}}$ However, the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ satisfying $\text{\hspace{0.17em}}y=f\left(x\right)\text{\hspace{0.17em}}$ is a curve. The curve shown includes $\text{\hspace{0.17em}}\left(0,2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(6,1\right)\text{\hspace{0.17em}}$ because the curve passes through those points.

The vertical line test    can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See [link] .

Given a graph, use the vertical line test to determine if the graph represents a function.

1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.
2. If there is any such line, determine that the graph does not represent a function.

## Applying the vertical line test

Which of the graphs in [link] represent(s) a function $\text{\hspace{0.17em}}y=f\left(x\right)?$

If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of [link] . From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, at most x -values, a vertical line would intersect the graph at more than one point, as shown in [link] .

Does the graph in [link] represent a function?

yes

## Using the horizontal line test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test    . Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.

1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.
2. If there is any such line, determine that the function is not one-to-one.

## Applying the horizontal line test

Consider the functions shown in [link] (a) and [link] (b) . Are either of the functions one-to-one?

The function in [link] (a) is not one-to-one. The horizontal line shown in [link] intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.)

The function in [link] (b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.

Is the graph shown in [link] one-to-one?

No, because it does not pass the horizontal line test.

## Identifying basic toolkit functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as the input variable and $\text{\hspace{0.17em}}y=f\left(x\right)\text{\hspace{0.17em}}$ as the output variable.

#### Questions & Answers

how can are find the domain and range of a relations
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
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim