Functions Final Exam

By mathematical convention, coordinates are always written in alphabetical order as $(x, y)$. This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.

A relation is a function if and only if each input value is associated with exactly one output value. If a single input has multiple different outputs, it is not a function.

To find where a function crosses the $y$-axis, calculate $f(0)$ by replacing $x$ with $0$ in the function formula. The resulting point on the graph will be $(0, f(0))$.

If you input a value outside the domain, the function is undefined for that value. For example, in $f(x) = \frac{1}{x}$, inputting $x = 0$ results in division by zero, which has no defined mathematical value.

To find the domain from a graph, look at the graph's horizontal extent along the $x$-axis. Find the leftmost and rightmost points of the graph, taking note of whether the endpoints are solid (included) or open circles (excluded).

The domain is the set of all valid input values (usually $x$) that you can feed into a function, while the range is the set of all output values (usually $y$) that the function produces as a result.

To find the range from a graph, look at the graph's vertical extent along the $y$-axis. Find the lowest and highest points of the graph, taking note of whether these endpoints are included (solid) or excluded (open circles).

Intuitively, it means you can draw the function's graph without lifting your pencil. Formally, a function must be defined at the point, and the graph must connect smoothly without any gaps, jumps, or holes.

Look for inputs that make the function undefined (like division by zero). For example, $f(x) = \frac{x^2 - 16}{x - 4}$ is discontinuous at $x = 4$ because you cannot divide by zero, creating a hole in the graph.

For linear functions $f(x) = mx + b$, the function increases if the slope $m > 0$ and decreases if $m < 0$. For quadratic functions $f(x) = a(x-h)^2 + k$, check the sign of $a$: if $a > 0$, the parabola opens up, so it decreases for $x < h$ and increases for $x > h$.

Although we look at the vertical rise or fall (y-values) to determine *what* the graph does, the intervals must specify *where* it happens horizontally. By convention, intervals of increase/decrease partition the domain of the function, which is represented by the $x$-axis.