1.1 Functions and Function Notation

Functions and Their Representations

Functions describe a specific kind of relationship: every input gets exactly one output. That single rule is what separates a function from a general relation. Once you're comfortable identifying and working with functions in all their forms, you'll have the foundation for nearly everything else in pre-calc.

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Classification of Function Representations

A function can show up in four different forms, and you need to move fluently between all of them.

• Equations express the input-output relationship with mathematical symbols. For example, $f(x) = 3x - 5$ tells you exactly how to compute the output for any input.
• Tables organize input-output pairs in rows or columns. They're useful when you have discrete data points rather than a continuous rule.
• Graphs plot inputs on the horizontal axis and outputs on the vertical axis, giving you a visual picture of the relationship. You can spot behavior like increasing/decreasing trends or symmetry at a glance.
• Verbal descriptions explain the relationship in words, often tied to a real-world context. Something like "the cost is $15 per hour plus a $20 flat fee" is a verbal description of $C(h) = 15h + 20$.

The same function can appear in any of these forms. A common exam task is translating between them, so practice going from a table to an equation, or from a verbal description to a graph.

Interpretation of Function Outputs

Interpreting a function's output means understanding what the number actually tells you in context. If $P(t) = 500$ where $P$ models profit in dollars and $t$ is months, then the output means $500 of profit at that particular month.

Always pay attention to units. The output might represent dollars, meters, degrees, people, or something else entirely. Stating the units is what turns a bare number into a meaningful answer. On problems that ask you to "interpret," a complete response connects the numerical value back to the real-world quantity it represents.

One-to-One vs. Many-to-One Functions

Both one-to-one and many-to-one relationships qualify as functions (every input still has exactly one output). The difference is about whether outputs repeat.

• One-to-one functions pair each input with a unique output. No two different inputs ever produce the same output. For example, $f(x) = 2x + 1$ is one-to-one because different $x$-values always give different results.
• Many-to-one functions allow multiple inputs to share the same output. For example, $f(x) = x^2$ is many-to-one because $f(2) = 4$ and $f(-2) = 4$. Two different inputs map to the same output.

Why does this matter? One-to-one functions are the only functions that have true inverses. You can check whether a function is one-to-one using the horizontal line test: if any horizontal line crosses the graph more than once, the function is not one-to-one.

Vertical Line Test for Functions

The vertical line test tells you whether a graph represents a function at all. The logic comes straight from the definition: each input ($x$-value) can have at most one output ($y$-value).

1. Look at (or imagine drawing) vertical lines across the entire graph.
2. If every vertical line hits the graph at one point or fewer, the relation is a function.
3. If any vertical line hits the graph at two or more points, the relation is not a function, because that $x$-value would have multiple outputs.

A circle, for instance, fails the vertical line test. A parabola opening upward passes it.

Graphs of Common Functions

You should recognize these on sight and know their key features.

Linear functions have the form $y = mx + b$ and produce a straight line.

• $m$ is the slope (rate of change): rise over run.
• $b$ is the $y$-intercept, the output when $x = 0$.
• Example: $y = 2x + 3$ has slope 2 and crosses the $y$-axis at $(0, 3)$.

Quadratic functions have the form $y = ax^2 + bx + c$ and produce a parabola.

• The vertex is the highest or lowest point, depending on the sign of $a$ (opens up if $a > 0$, down if $a < 0$).
• The axis of symmetry is the vertical line through the vertex, found by $x = \frac{-b}{2a}$.
• Example: $y = -x^2 + 4x - 3$ opens downward and has its vertex at $(2, 1)$.

Exponential functions have the form $y = a \cdot b^x$, where the variable is in the exponent.

• $a$ is the initial value (output when $x = 0$).
• $b$ is the base: if $b > 1$, the function models growth; if $0 < b < 1$, it models decay.
• Example: $y = 2 \cdot 3^x$ starts at 2 and triples with each unit increase in $x$.

Piecewise functions use different rules on different intervals of the input. You'll see them written with a brace grouping two or more equations, each with a specified domain. When graphing, pay close attention to open vs. closed circles at the boundary points.

Advanced Function Concepts

• Inverse functions reverse the input-output relationship. If $f(3) = 7$, then $f^{-1}(7) = 3$. Only one-to-one functions have inverses that are also functions.
• Composition of functions feeds the output of one function into another. The notation $f(g(x))$ means "evaluate $g$ first, then plug that result into $f$." Work from the inside out.
• Functions with multiple inputs take more than one variable to determine a single output. You'll encounter these more in later courses, but the core idea (each combination of inputs yields exactly one output) stays the same.