FUNCTIONS 02
Function Notation
Function notation tells us which function is being used, what input is supplied, and which output is produced.
The main idea
If a function is named \(f\) and the input is \(x\), the corresponding output is written
\[\boxed{f(x)}\]and read “\(f\) of \(x\)”. The brackets identify the input; they do not mean multiplication.
1. Reading the notation
For
\[f(x)=2x+3,\]the notation has three distinct parts.
function name
input
output
2. Evaluating a function
To evaluate a function, substitute the chosen input wherever the input variable appears.
input \(x=4\)
\(f(4)=2(4)+3=11\)
3. The input can be a number, variable or expression
The brackets can contain any allowed input, not only a number.
If \(f(x)=x^2+1\), then
\(f(a)=a^2+1\)
\(f(x+h)=(x+h)^2+1\)
4. Different function names
The letter \(f\) is only a label. Different functions can be given different names.
\(g(x)=2x+1\)
\(h(t)=e^{-t}\)
5. \(y\) and \(f(x)\)
The equations
\[y=2x+1\qquad\text{and}\qquad f(x)=2x+1\]can describe the same relationship. Function notation makes the dependence on the input explicit and is especially useful when several functions appear together.
or
\(x\mapsto f(x)\)
same output relationship
6. Function notation is not multiplication
\(f(x)\) does not mean \(f\times x\). It means “the value produced by function \(f\) at input \(x\)”.
means
input \(2\) gives output \(7\)
not \(f\times2\)
7. Comparing two function values
For \(f(x)=x^2\), compare the outputs at \(x=2\) and \(x=5\):
\[f(2)=4,\qquad f(5)=25.\]The change in output is \(25-4=21\).
8. Nested notation
If one function is used inside another, notation records the order clearly.
If \(f(x)=x^2\) and \(g(x)=x+1\), then
\[f(g(2))=f(3)=9.\]so
\(f(g(2))=9\)
9. Function notation with other independent variables
The input variable does not have to be \(x\). In applications, the symbol often reflects the quantity being modelled.
\(T(x)\): temperature at position \(x\)
\(V(r)\): volume as a function of radius \(r\)
10. The notation that leads to calculus
If the input changes from \(x\) to \(x+\Delta x\), the corresponding output changes from \(f(x)\) to \(f(x+\Delta x)\).
Therefore
\[\Delta y=f(x+\Delta x)-f(x),\]and the average rate of change is
\[\boxed{\frac{f(x+\Delta x)-f(x)}{\Delta x}}.\]↓
\(f(x)\to f(x+\Delta x)\)
↓
difference quotient
\(f(x)\) names the output of function \(f\) at input \(x\). The input can be a number, variable or expression. Function notation is not multiplication, and it becomes especially powerful when comparing outputs, composing functions and forming the difference quotient used in calculus.
Why this matters for calculus
Calculus repeatedly compares nearby function values. Expressions such as \(f(x+h)\), \(f(x+\Delta x)-f(x)\), and later \(f'(x)\), are meaningful only when function notation is read fluently.