FUNCTIONS 01
What Is a Function?
A function is a rule that assigns each allowed input exactly one output.
The main idea
A function describes dependence: when an input is chosen, the function determines the corresponding output.
\[\boxed{\text{input}\;x\longrightarrow\text{function }f\longrightarrow\text{output }f(x)}\]The essential condition is simple: one allowed input cannot be assigned two different outputs by the same function.
1. A function as a machine
Suppose
\[f(x)=2x+1.\]The rule says: multiply the input by \(2\), then add \(1\).
↓ multiply by 2
\(6\)
↓ add 1
output \(7\)
\(f(3)=7\)
2. A function is a mapping
A mapping diagram should show exactly where each input goes.
For \(f(x)=x^2\), both \(-2\) and \(2\) map to \(4\), while \(0\) maps to \(0\).
3. Exactly one output for each input
A relation is a function only when each allowed input has exactly one output.
An input cannot simultaneously produce two different outputs.
not a function:
\(2\mapsto4\) and \(2\mapsto7\)
4. Different inputs may share an output
The rule does not require different inputs to have different outputs.
For \(f(x)=x^2\),
\[f(-2)=4,\qquad f(2)=4.\]The graph is symmetric about the \(y\)-axis, so opposite inputs have the same output.
5. A function is more than a formula
A formula is one way to describe a function, but a function can also be represented by a table, graph, mapping diagram or verbal rule.
table: \((1,1),(2,4),(3,9)\)
graph: points \((x,x^2)\)
words: “square the input”
6. The graph of a function
Every point on the graph has the form
\[(x,f(x)).\]The horizontal coordinate is the input and the vertical coordinate is its output.
7. The vertical-line test
A graph represents \(y\) as a function of \(x\) if every vertical line meets the graph at most once.
If one vertical line meets it twice, the same \(x\)-value has two outputs, so it is not a function of \(x\).
8. Input and output are not interchangeable
For \(f(x)=x^2\), the statement \(f(3)=9\) means that input \(3\) produces output \(9\).
It does not automatically mean that input \(9\) produces output \(3\): in fact \(f(9)=81\).
but
\(9\xrightarrow{f}81\)
9. Functions describe dependence in models
Functions are used whenever one quantity depends on another.
If population depends on time, we may write \(P(t)\). If temperature depends on position, we may write \(T(x)\).
position \(x\to T(x)\): temperature
dose \(d\to R(d)\): response
10. The bridge to calculus
Once a quantity is represented by a function, we can ask how its output changes when its input changes.
\[x\to x+\Delta x\qquad\Rightarrow\qquad f(x)\to f(x+\Delta x).\]The resulting change in output is
\[\Delta f=f(x+\Delta x)-f(x).\]\(x\to x+\Delta x\)
↓
output changes
\(f(x)\to f(x+\Delta x)\)
↓
calculus studies the rate
A function assigns each allowed input exactly one output. A mapping diagram should show an arrow from every input to its output. Different inputs may share an output. On a graph, every allowed \(x\)-value has only one corresponding \(y\)-value.
Why this matters for calculus
Derivatives and integrals act on functions. Before asking how quickly something changes, we must first understand which quantity depends on which input and how that dependence is represented.