You can think of a function like a machine. You input something into the machine and the function will output something.
For example, the function may be a machine that triples every number.
Another example may be the function \(x^2 +1\). This function takes every input and squares it, then adds it by 1.
A function can be named anything, the most common function names are \(f\) and \(g\). There is an input inside the function that can also be denoted by any letter, but most commonly it is denoted by \(x\). Thus, \[f(x)\]
means that the name of the function is \(f\) and the input is \(x\). The output is what it is equal to.
\[f(x)=x^2\]
is read "f of x equals x squared" where \(x^2\) is the output.
Another example \[g(A)=A+2\]
is a function that is named \(g\), with input \(A\), and the output adds all inputs by 2 to get \(A+2\).
There are special functions with a specific names for example \(sin(x)\) and \(ln(x)\).
All possible values (or elements) that can be inputted into a function belong to what we call the domain. Whereas all possible elements that can be outputted belong to what we call the range or codomain.
Not every equation can be a function, a function must have the following:
In other words, one element cannot lead to two outputs in the range.
This is a function because every element in the domain X has one unique output, and every element in domain X is used. No input is left out.
This is not a function because the element 2 in the domain has more than one output and not every element in the domain is used (e.g., 3 and 4 have no outputs).
For \(f(x) = 2x^2 -1\), \(g(y) = 9y +3\), \(h(x) = \sqrt{2x-1}\), find
\(f(3) \) means that for the function named \(f\), which is represents \(2x^2 -1\), the input is 3 or \(x=3\). In other words, substitute \(x=3\) into \(2x^2 -1\) and find the output.
\(f(3) = 2(\textbf{3})^2 -1 = 2(\textbf{9}) - 1 = 17\)
\(g(-3) \) means to substitute \(y=-3\) into \(g(x) = 9y +3\)
\(g(-3) = 9(\textbf{-3}) +3 = 30\)
\(h(A) \) means to substitute \(x=A\) into \(h(x) = \sqrt{2x-1}\)
\(h(A) = \sqrt{2A-1}\)
\(f(x+1) \) means to substitute \(x=x+1\) into \(f(x) = 2x^2-1\)
\(f(x+1) =2(\textbf{x+1})^2-1\)
\(g(y^2) \) means to substitute \(y=y^2\) into \(g(y) = 9y +3\)
\(g(y^2) = 9(\textbf{y^2}) +3\)
\(h(9x+3) \) means to substitute \(x=9x+3\) into \(h(x) = \sqrt{2x-1}\)
\(h(A) = \sqrt{2(\textbf{9x+3})-1} = \sqrt{\textbf{18x+6}-1} = \sqrt{18x+5}\)
