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.
A 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.
The most common 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:
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.
Let $f(x) = \sqrt{x + 4}$. The value of $f(x)$ at $x = 5$ is $f(5) = \sqrt{4 + 5} = 3$.
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.
The domain is often determined by considering restrictions on the input values:
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.
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