A function can be defined via a graph. Each dot on a graph is highlighting a point
and the coordinates of this point define a pair in the function. We can use a
graph to estimate function values, but we cannot escape the approximation
inherent to drawing. To communicate about exactness, some functions have an
algebraic description of the pairings. We call this algebraic tool a formula or an
equation. Not all functions have formulas with which we can calculate.
But when they do, that’s what we want to use! Our goal is to be exact.
When functions have formulas, then there is an operation manual to follow.
It is not that simple!!!!
It never is.
We have a beginning...
A formula for a function is an algebraic expression involving the domain number, that produces the function value at the domain number.
In function notation, \(f(d)\), the symbol inside the parentheses is called the variable. It represents all of the domain values.
...but, that is just the beginning.
We will begin thinking about function notation in the form
Function Name(Variable)
However, we will quickly move to forms like
Function Name(Expression)
Function notation is a sophisticated way to communicate. It will evolve
as we wish to communicate more and more information about a function.
In function notation, the inside of the parentheses may hold just the variable. In
that case, the variable for the formula is representing the domain values.
Or, the inside of the parenthses may hold an expression, which involves the variable.
In this case, the variable is not representing the domain values. The expression
is.
The variable does not always represent domain numbers.
The inside of the parentheses ALWAYS represents the domain
numbers.
Sometimes the inside of the parentheses is just the variable and so the variable is
representing the domain numbers.
Sometimes not.
The function \(P\) is defined as follows.
\(P(k) = 3k - 2\)
domain = \([-2, 6)\)
range = \([-8, 16)\)
Here, the inside of the parentheses in the function notation is \(k\). That tells us that \(k\) represents the domain numbers. The variable \(k\) represents the numbers \([-2, 6)\).
The function \(R\) is defined as follows.
\(R(k + 5) = 3k - 2\)
domain = \([-2, 6)\)
Here, the inside of the parentheses in the function notation is \(k + 5\). That is not the
variable.
\(k\) is the variable. That is not what is inside the parentheses.
The variable \(k\) DOES NOT represent the numbers \([-2, 6)\), which is the domain.
The expression \(k + 2\) represents the domain numbers. The expression \(k + 5\) represents the
domain numbers \([-2, 6)\).
\(k\) represents the numbers that if you added \(5\) to them, then you would get the domain
numbers - you would get the numbers \([-2, 6)\).
\(k\) represents the number \([-7, 1)\).
When using function notation and formulas, the inside of the parentheses is much
more important than the variable.
Q: Why not just keep it simple and just always have the inside of the parentheses be
the variable?
A: Because, we want to use functions to describe the world and the world is not that
simple.
The function \(R\) is defined as follows,
\(R(k + 5) = 3k - 2\)
domain = \([-2, 6)\)
Let’s define a new function called \(B\). The definition of \(B\) will use \(R\).
Let’s define \(B\) as follows,
The values of \(B\) are found as values of \(R\). We use the word induced for these situations.
We don’t have a direct way to calculate values of \(B\). Instead, \(R\) will induce/cause the
values of \(B\).
Evaluate \(B(-1)\).
step 1) We need \(B(-1) = B(t - 1)\), which makes \(t = \answer {0}\).
step 2) If \(t = 0\), then replace all occurrences of \(t\) with \(0\).
step 3) We have a way to evaluate \(R(4)\).
We need \(R(4) = R(k + 5)\), which makes \(k = \answer {-1}\).
step 4) Use the formula for \(R\).
All together
This is the type of reasoning we need to work with functions.
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more examples can be found by following this link
More Examples of Formulas