# 1.1 Functions and function notation  (Page 7/21)

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Using [link] , solve $\text{\hspace{0.17em}}f\left(x\right)=1.$

$x=0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$

## Determining whether a function is one-to-one

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in [link] at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of$1000.

However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in [link] .

A 4.0
B 3.0
C 2.0
D 1.0

This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.

To visualize this concept, let’s look again at the two simple functions sketched in [link] (a) and [link] (b) . The function in part (a) shows a relationship that is not a one-to-one function because inputs $\text{\hspace{0.17em}}q\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ both give output $\text{\hspace{0.17em}}n.\text{\hspace{0.17em}}$ The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

## One-to-one function

A one-to-one function    is a function in which each output value corresponds to exactly one input value.

## Determining whether a relationship is a one-to-one function

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

A circle of radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ has a unique area measure given by $\text{\hspace{0.17em}}A=\pi {r}^{2},$ so for any input, $\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ there is only one output, $A.$ The area is a function of radius $\text{\hspace{0.17em}}r.$

If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is given by the formula $\text{\hspace{0.17em}}A=\pi {r}^{2}.\text{\hspace{0.17em}}$ Because areas and radii are positive numbers, there is exactly one solution: $\sqrt{\frac{A}{\pi }}.$ So the area of a circle is a one-to-one function of the circle’s radius.

1. Is a balance a function of the bank account number?
2. Is a bank account number a function of the balance?
3. Is a balance a one-to-one function of the bank account number?

a. yes, because each bank account has a single balance at any given time; b. no, because several bank account numbers may have the same balance; c. no, because the same output may correspond to more than one input.

Evaluate the following:

1. If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
2. If so, is the function one-to-one?
2. No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

## Using the vertical line test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.

how to understand calculus?
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
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
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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
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?