Key Concepts of Function Notation

Why This Matters

Function notation is the language algebra uses to describe relationships between quantities—and it's everywhere on your exams. When you see $f(x)$, you're not just looking at a fancy way to write an equation; you're working with a powerful tool that lets you evaluate specific values, combine operations, and analyze how quantities change together. Mastering function notation connects directly to graphing, solving equations, modeling real-world scenarios, and understanding more advanced topics like composition and inverses.

Here's the key insight: you're being tested on whether you understand functions as input-output machines with specific rules and behaviors. Don't just memorize that $f(x) = 2x + 3$—know why we use this notation, how to work with it flexibly, and what it reveals about the relationship between variables. Once you see functions as a system of connected concepts, everything from domain restrictions to inverse operations clicks into place.

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Understanding the Function Framework

Before you can work with functions, you need to grasp what makes something a function in the first place. The core rule is simple: every input gets exactly one output—no exceptions.

Definition of a Function

• One input, one output—this is the fundamental rule that separates functions from other relations
• Multiple representations exist for the same function: equations, graphs, tables, or even verbal descriptions
• The vertical line test determines if a graph represents a function; if any vertical line crosses the graph more than once, it's not a function

Function Notation $f(x)$

• $f(x)$ names both the function and its input—the letter $f$ is the function's name, and $x$ is the variable you're plugging in
• Parentheses don't mean multiplication here; $f(x)$ reads as "f of x," meaning the output when $x$ is the input
• Substitution becomes straightforward: if $f(x) = 2x + 3$, then $f(2) = 2(2) + 3 = 7$

Input and Output of Functions

• Input values are what you control—these are the $x$-values you substitute into the function
• Output values are what you get—the result after the function "processes" your input, written as $f(x)$ or $y$
• The relationship is deterministic: the same input always produces the same output in a given function

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Boundaries and Behavior

Understanding where a function works and what values it produces is essential for solving problems correctly. Domain and range define the function's limits.

Domain and Range

• Domain is all possible inputs—ask yourself, "What $x$-values can I plug in without breaking the math?"
• Range is all possible outputs—the set of $f(x)$ values the function can actually produce
• Common domain restrictions include avoiding division by zero and square roots of negative numbers

Evaluating Functions

• Substitute the input everywhere you see $x$—replace every instance, not just one
• Follow order of operations carefully; parentheses around your substituted value prevent errors
• Evaluation answers the question "What does this function output when I input this specific value?"

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Visualizing Functions

Graphs transform abstract equations into visual patterns you can analyze at a glance. The coordinate plane maps inputs to outputs as points.

Graphing Functions

• The x-axis represents inputs, the y-axis represents outputs—each point $(x, y)$ shows an input-output pair
• Plot points by evaluating the function at several $x$-values, then connect them appropriately
• Shape reveals behavior: straight lines indicate constant change, curves indicate variable change

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Types of Functions

Different function types have distinct forms and behaviors. Recognizing the equation form tells you what the graph will look like.

Linear Functions

• Standard form is $f(x) = mx + b$ where $m$ is the slope and $b$ is the y-intercept
• Constant rate of change—the slope $m$ tells you how much $f(x)$ changes for every 1-unit increase in $x$
• Graph is always a straight line, making linear functions the simplest to graph and analyze

Quadratic Functions

• Standard form is $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants
• Graph is a parabola—opens upward when $a > 0$, downward when $a < 0$
• Variable rate of change means the function speeds up or slows down, unlike linear functions

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Combining and Reversing Functions

Advanced function work involves using functions together or undoing their operations. These concepts appear frequently in multi-step problems.

Function Composition

• Notation $(f \circ g)(x) = f(g(x))$ means "plug $g(x)$ into $f$"—work from the inside out
• Order matters: $(f \circ g)(x)$ is usually different from $(g \circ f)(x)$
• Chain the operations by first evaluating the inner function, then using that result as input for the outer function

Inverse Functions

• An inverse "undoes" the original function—if $f(x)$ turns 2 into 7, then $f^{-1}(x)$ turns 7 back into 2
• Notation $f^{-1}(x)$ does not mean $\frac{1}{f(x)}$; the superscript indicates inverse, not reciprocal
• Finding inverses involves swapping $x$ and $y$, then solving for $y$

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Quick Reference Table

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Self-Check Questions

1. If $f(x) = 3x - 1$ and $g(x) = x^2$, what is $(f \circ g)(2)$? What is $(g \circ f)(2)$? Why are they different?

2. A function has the equation $f(x) = \frac{1}{x-4}$. What value must be excluded from the domain, and why?

3. Compare and contrast linear and quadratic functions: how do their rates of change differ, and how can you identify each from an equation?

4. If $f(3) = 10$, what can you conclude about $f^{-1}(10)$? Explain the relationship between a function and its inverse.

5. Given a table of values where the input 5 produces two different outputs, explain why this relation is not a function using the definition you learned.