Functions and Their Notation
What is a function?
A function is a mathematical object that describes the relationship between an input and an output. Think of it like a box that takes in some input value $x$ and spits out an output value $f(x)$.
Let $f(x) = 2x + 3$. Here, $f$ is the function name, $x$ is the input, and $2x + 3$ is the rule that defines the function.
A function performs an operation on the input to turn it into an output. For example, the function $f(x) = x^2 + 4$ squares the input, then adds 4.
It's important that the function only has one output for each input. If a single input can map to multiple outputs, for example $f(x) = \pm \sqrt{x}$, then that is not a function, simply a relation.
NoteA function can still map different inputs to a single output, for example $f(x) = x^2$ maps both $-2$ and $2$ to $4$. As long as it doesn't yield two different possible outputs for the same input, it can still be considered a function.
Function Notation
The most notation for a function is $f(x)$, where $f$ is the name of the function and $x$ is the input variable. However, other letters can be used depending on the context:
- $v(t)$ might represent velocity as a function of time
- $C(n)$ could denote cost as a function of the number of items
When working with multiple functions, exams may use different letters to distinguish them, such as $f(x)$, $g(x)$, and $h(x)$.
Occasionally, the following notation will also be used:
$$f : x \to y$$
where $y$ is an expression. This means the same thing as $f(x) = y$, where the function maps the input $x$ to the output $y$.
When evaluating a function at a specific number, we use the notation $f(n)$ where $n$ is the number you are evaluating the function at.
ExampleLet $f(x) = \sqrt{x + 4}$. The value of $f(x)$ at $x = 5$ is $f(5) = \sqrt{4 + 5} = 3$.
Domain and Range
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) that the function can produce.
Domain
The domain is often determined by considering restrictions on the input values:
- For rational functions, exclude values that make the denominator zero.
- For square root functions, ensure the expression under the square root is non-negative.
- For logarithmic functions, the argument must be positive.
For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except 2, because when $x = 2$, the denominator becomes zero.
For $f(x) = \sqrt{1 - x}$, the domain is all real numbers less than or equal to 1, because if $x > 1$, the expression under the square root is negative.
For $f(x) = \log(2x - 1)$, the domain is all real numbers greater than $\frac12$, because if $x \leq \frac12$, the argument is negative or zero.
For some functions, the domain can also be restricted. This means that the function is essentially "cut off" at all points not in the domain defined.
Example$f(x) = x^2, x\geq 0$ means the function is only defined for values of $x$ greater than or equal to $0$.
Common MistakeStudents often forget to check whether the range changes as a result of a restricted domain. For example, the range of $f(x) = \cos x$ is $-1 \leq x \leq 1$, but the range of $f(x) = \cos x, -\frac\pi2 \leq x \leq \frac\pi2$ is $0 \leq x \leq 1$.
Range
The range can often be determined by analyzing the function's behavior:
- For polynomial functions, consider end behavior and turning points.
- For trigonometric functions, recall their periodic nature.
- For exponential functions, consider asymptotic behavior.
For $g(x) = x^2$, the range is all non-negative real numbers, because a square is always non-negative.
For $g(x) = \sin x$, the range is all real numbers between -1 and 1.
For $g(x) = e^x$, the range is all positive real numbers, because $e^x$ cannot equal a negative number or zero.
Notating domain and range
Domains and ranges can be written using inequalities. For example:
- The domain of $f(x) = \sqrt{x}$ is $x\geq 0$.
- The range of $f(x) = \sin x$ is $-1 \leq f(x) \leq 1$.
