Functions - Function notation and graphing

Function Definition and Vertical Line Test: A function is a relation where every input $x$ has exactly one output $y$. Visually, a graph represents a function if any vertical line drawn through it intersects the curve at most once. If a vertical line hits the graph twice, it is a relation but not a function.

Function Notation: We use $f(x)$ to denote the output of a function for a given input $x$. For example, if $f(x) = 2x + 3$, then $f(4)$ means substituting $4$ for $x$. On a graph, the $x$-coordinate is the input, and the $y$-coordinate ($f(x)$) represents the vertical height of the point.

Domain and Range: The domain is the set of all possible input values ($x$-axis), while the range is the set of all possible output values ($y$-axis). On a graph, the domain is the horizontal 'spread' from left to right, and the range is the vertical 'spread' from the lowest point to the highest point.

Linear Functions: These take the form $f(x) = mx + c$. The graph is a straight line where $m$ is the gradient (slope) and $c$ is the $y$-intercept. A positive $m$ results in a line sloping upwards to the right, while a negative $m$ slopes downwards.

Quadratic Functions: Written as $f(x) = ax^2 + bx + c$, these form a symmetrical curve called a parabola. If $a > 0$, the parabola opens upwards (U-shape) with a minimum point; if $a < 0$, it opens downwards (n-shape) with a maximum point called the vertex.

Intercepts on a Graph: The $y$-intercept occurs where the graph crosses the vertical axis, found by calculating $f(0)$. The $x$-intercepts (or zeros) are where the graph crosses the horizontal axis, found by solving the equation $f(x) = 0$.

Composite Functions: This involves placing one function inside another, denoted as $(f \circ g)(x)$ or $f(g(x))$. Visually, this means the output of the 'inner' function $g$ becomes the input for the 'outer' function $f$.