I read the meaning of function notation but i still don't understand why are they so important.
during an art contest at your school, you and a classmate each won blue ribbons for one third of the pieces you entered in the contest. You won two blue ribbons and your classmate won three blue ribbons. Explain how this could be.
during an art contest at your school, you and a classmate each won blue ribbons for one third of the pieces you entered in the contest. You won two blue ribbons and your classmate won three blue ribbons. Explain how this could be.
during an art contest at your school, you and a classmate each won blue ribbons for one third of the pieces you entered in the contest. You won two blue ribbons and your classmate won three blue ribbons. Explain how this could be
during an art contest at your school, you and a classmate each won blue ribbons for one third of the pieces you entered in the contest. You won two blue ribbons and your classmate won three blue ribbons. Explain how this could be
what does your problems have to do with function notation?
Efficient notations in general are important. To appreciate this, compare to how people did math in the old days
Tartaglia explained in a poem how to solve third degree equations of the form:
x^3 + p x + q = 0
By rescaling x you can make p equal to one, so you only have to consider the case x^3 + x + q = 0.:
"When the cube and things together
Are equal to some discreet number,
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing
In the second of these acts,
When the cube remains alone,
You will observe these other agreements:
You will at once divide the number into two parts
So that the one times the other produces clearly
The cube of the third of the things exactly.
Then of these two parts, as a habitual rule,
You will take the cube roots added together,
And this sum will be your thought.
The third of these calculations of ours
Is solved with the second if you take good care,
As in their nature they are almost matched.
These things I found, and not with sluggish steps,
In the year one thousand five hundred, four and thirty.
With foundations strong and sturdy
In the city girdled by the sea."
the reason why function notations are important is because with equations you have no direction, so you change no direction, only with inequalities. Is this correct.
Maybe you have to discuss this with your teacher in class. My opinion is that it is just a convenient notation. You can do without it, but then it becomes cumbersome to formulate things.
It's just the same with language. Ani,als communicate with each other using very simple noises that directly refer to actual things. We humans can use more abstract concepts. We can refer to things that will happen, things that could have happened etc.
In maths you can also see different levels of abstraction. At first you learn to work with numbers. That's the lowest level of abstraction. Then you learn to use variables that can take any value. If you say x + x = 2x then this is a statement that is true whatever value x takes.
You can consider functions of x like x^2. We go to the next higher level of abstraction by considering a general function f(x) without specifying it completely.
scientific notation
i need help with fractions what do i do?
To understand the importance of function notation, let's first review what function notation is. In mathematics, a function is a relation between a set of inputs (called the domain) and a set of outputs (called the range). Function notation is a way to represent a function using symbols and conventions. It typically involves using a letter (often "f", "g", or "h") and parentheses to show the relationship between the input and output.
Now, why are function notations so important? Here are a few reasons:
1. Clarity and Precision: Function notation provides a concise and clear way to represent mathematical relationships. It helps avoid ambiguity and makes it easier to communicate and understand mathematical ideas.
2. Flexibility: Function notations allow us to define and work with a wide variety of functions. We can represent simple linear functions, complex polynomial functions, exponential functions, trigonometric functions, and many more using function notation.
3. Analysis and Manipulation: Function notation enables us to analyze and manipulate functions in various ways. We can find the domain and range of a function, evaluate it at specific input values, find its inverse, and perform operations like composition and transformation of functions.
4. Higher-level Mathematics: Function notation is fundamental in higher-level mathematics, such as calculus and advanced algebra. It facilitates the study of limits, derivatives, integrals, sequences, series, and other important concepts.
Now, let's move on to the art contest problem you mentioned. In this scenario, you and your classmate each won blue ribbons for one-third of the pieces you entered. You won two blue ribbons, and your classmate won three blue ribbons. To explain this, we can use algebraic reasoning.
Let's say you entered 'x' pieces into the contest. According to the given information, you won blue ribbons for one-third of your entries. Therefore, the number of blue ribbons you won can be expressed as (1/3)x.
Similarly, your classmate entered 'y' pieces into the contest. According to the given information, your classmate won blue ribbons for one-third of their entries. Therefore, the number of blue ribbons your classmate won can be expressed as (1/3)y.
From the problem, we know that you won two blue ribbons and your classmate won three blue ribbons. So, we can set up two equations using the expressions we derived:
(1/3)x = 2 (You won two blue ribbons)
(1/3)y = 3 (Your classmate won three blue ribbons)
Now, we can solve these equations to find the values of 'x' and 'y'.
Multiplying both sides of the first equation by 3, we get:
x = 6 (You entered 6 pieces into the contest)
Multiplying both sides of the second equation by 3, we get:
y = 9 (Your classmate entered 9 pieces into the contest)
So, you won two blue ribbons out of the six pieces you entered, while your classmate won three blue ribbons out of the nine pieces they entered. This satisfies the condition that you both won blue ribbons for one-third of your entries.
In summary, function notation is important in mathematics for its clarity, precision, flexibility, and its role in higher-level mathematical concepts. It allows us to represent and analyze mathematical relationships and solve problems efficiently.