Think of a function as a rule, like a recipe. You give it an input (an ingredient), and it gives you exactly one output (a finished dish). In math, we usually work with numbers. You put a number in, the function performs an operation on it, and you get exactly one number out.

This relationship is often written as $f(x)$, which you read as "f of x." The letter $f$ is the name of the function (the rule), and $x$ is the input. For example, if we have the function $f(x) = x + 5$, it tells us to take any input $x$ and add 5 to it. If we plug in $x = 3$, we get $f(3) = 3 + 5 = 8$. The input is 3, and the output is 8.

Every function has two important sets of values associated with it: the domain and the range.

Imagine a function that squares any number you give it, $f(x) = x^2$. You can put any real number into this function, so its domain is "all real numbers." However, the output will always be zero or positive. You can't square a number and get a negative result. So, its range is "all real numbers greater than or equal to 0."

A graph is a visual representation of a function. It plots every input-output pair $(x, y)$ as a point on a coordinate plane. The x-axis represents the domain (inputs), and the y-axis represents the range (outputs). By connecting these points, we can see the shape of the function and understand its behavior.

Let's look at a few common types of functions.

Linear Functions: These create straight lines on a graph. Their rule is $f(x) = mx + b$. Here, $m$ represents the slope (how steep the line is) and $b$ is the y-intercept (where the line crosses the y-axis).

Polynomial Functions: These are built from variables raised to positive integer powers, like $f(x) = x^2$ (a parabola) or $f(x) = x^3 - 2x + 1$ (a cubic function).

Exponential and Logarithmic Functions: Exponential functions, like $f(x) = 2^x$, show rapid growth or decay. Logarithmic functions, like $f(x) = \log_2(x)$, are their inverses and grow very slowly.

Once you know the shape of a basic function, like $f(x) = x^2$, you can create many variations by transforming it. These transformations shift, stretch, or reflect the graph.

• Vertical Shifts: Adding a constant outside the function moves the graph up or down. $f(x) + c$ shifts the graph up by $c$ units. $f(x) - c$ shifts it down.
• Horizontal Shifts: Adding a constant inside the function moves the graph left or right. $f(x - c)$ shifts the graph to the right by $c$ units. $f(x + c)$ shifts it to the left.
• Reflections: A negative sign can reflect the graph. $-f(x)$ reflects it across the x-axis (flipping it upside down). $f(-x)$ reflects it across the y-axis (flipping it left to right).

Some functions have an inverse, which essentially undoes the original function's operation. If a function $f$ turns an input $a$ into an output $b$, its inverse, written as $f^{-1}$, will turn $b$ back into $a$. In other words, if $f(a) = b$, then $f^{-1}(b) = a$.

A classic example is the relationship between exponential and logarithmic functions. The function $f(x) = 2^x$ takes an input and uses it as a power. Its inverse, $f^{-1}(x) = \log_2(x)$, finds what power you need to raise 2 to in order to get $x$.

Graphically, the inverse of a function is its reflection across the diagonal line $y = x$.

Let's check your understanding of these core concepts.

Understanding functions and how they are represented visually is a critical first step for tackling calculus.