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Jan 3, 2026

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6 pages

Understanding Functions: Domains, Ranges, and Graphs

Sam

@sam.burchard

Function notation is a powerful mathematical language that helps you... Show more

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Introduction to Functions

A relation is a set of ordered pairs (x,y) that create graphs. The domain is the set of all x-values (inputs) of a relation, while the range is the set of all y-values (outputs). Think of domain as how "wide" the relation is and range as how "tall" it is.

A relation becomes a function when every element in the domain corresponds to exactly one value in the range. In other words, no x-value can be paired with multiple y-values. The vertical line test helps identify functions visually—if any vertical line intersects a graph more than once, it's not a function.

Some functions have an additional property called one-to-one (1-1), meaning every element in the range corresponds to exactly one element in the domain. Neither x nor y values repeat in a one-to-one function. The horizontal line test confirms this property—a function is one-to-one if no horizontal line crosses the graph more than once.

Remember: Not all functions are one-to-one, but all one-to-one relations must first be functions. Master the vertical and horizontal line tests to quickly identify function properties on graphs!

Function Notation Basics

Function notation uses expressions like f(x) (read as "f of x") to show that a relation is a function, where f(x) is equivalent to y. Different letters can be used for functions, like h(t) = t² - 5 or g(x) = x - 3, depending on the context.

When evaluating functions, you simply substitute the input value for the variable. For example, if f(x) = x² + 1, then f(3) = 3² + 1 = 10. This substitution process works with any input, including variables or expressions like f $x+1$ = $x+1$ ² + 1 = x² + 2x + 2.

Piecewise functions require careful attention to conditions. For instance, if f(x) equals 2x + 3 when x < -1 and equals 3 - x when x ≥ -1, you must check which condition your input satisfies before calculating. This explains why f(-5) = 2(-5) + 3 = -7, while f(-1) = 3 - (-1) = 4.

Pro Tip: When working with composite functions or function operations like f(g(x)), always work from the inside out, substituting one step at a time to avoid errors.

Finite Relations

Finite relations contain a limited number of ordered pairs, making them easier to analyze completely. When examining finite relations, we can list every ordered pair and precisely identify the domain and range.

To determine if a relation is a function, check whether each x-value (domain element) appears exactly once. For example, in the ordered pairs (1,4), (2,0), (3,5), each x-value (1, 2, and 3) appears exactly once, making this a function. Additionally, since each y-value (4, 0, and 5) appears exactly once, this is also a one-to-one function.

Not all functions are one-to-one, however. The relation with ordered pairs (1,6), (2,6), (3,6) is a function because each x-value appears once, but it's not one-to-one since the y-value 6 repeats. Relations like (1,4), (2,4), (2,6), (3,5) aren't functions at all because the x-value 2 is paired with multiple y-values.

Visual Check: Arrow diagrams provide a quick way to verify functions—if any domain element has multiple arrows pointing from it, the relation is not a function!

Function Applications and Operations

Functions can model real-world scenarios like business profits. When working with related functions, you can solve for unknown functions using algebra. For instance, if profit p(x) equals revenue r(x) minus cost c(x), you can rearrange the equation to find c(x) = r(x) - p(x).

When combining functions, follow the rules of algebra carefully. For example, to find the product f(x)·g(x), multiply the functions together, and to find the sum f(x)+g(x), add them. These operations create new functions that may have different properties than the original functions.

Simplifying complex function expressions requires careful step-by-step work. For an expression like f(x)·g(x) - $f(x)+g(x)$ , substitute the function definitions, distribute terms, combine like terms, and express the result in standard form. This methodical approach ensures you don't miss terms or make sign errors.

Application Alert: Functions model countless real-world scenarios—from profit calculations to physics equations. The ability to manipulate functions algebraically will serve you in many science and business applications!

Infinite Relations

Unlike finite relations, infinite relations contain unlimited ordered pairs, making them impossible to list completely. Instead, we describe their domains and ranges using set notation or interval notation like x ∈ ℝ (x is an element of all real numbers) or y ≥ -1.

The vertical line test still determines if an infinite relation is a function. If a relation passes this test, every x-value corresponds to exactly one y-value. Similarly, the horizontal line test checks if a function is one-to-one. For example, a linear function like y = x passes both tests and is therefore a one-to-one function.

Some common functions have specific domain and range patterns. For instance, square root functions like y = √ $x-2$ have restricted domains (x ≥ 2 in this case) and ranges (y ≥ 0). Absolute value functions like y = -|x+2| have full domains (x ∈ ℝ) but restricted ranges (y ≤ 0 here).

Notation Tip: The symbols "∈" means "is an element of" and ℝ represents all real numbers. These symbols help express infinite domains and ranges concisely!

Graphing Different Types of Functions

Linear functions like 2x - y = 6 $which can be rewritten as y = 2x - 6$ have both domain and range extending across all real numbers. These functions create straight lines when graphed and pass both the vertical and horizontal line tests, making them one-to-one functions.

Absolute value functions like y = -|x + 2| create V-shaped graphs that are flipped upside down. They have domains of all real numbers, but their ranges are restricted (y ≤ 0 in this case). These functions pass the vertical line test but fail the horizontal line test, so they're functions but not one-to-one.

Square root functions like y = √ $x - 2$ have restricted domains because you can't take the square root of negative numbers. For this function, the domain is x ≥ 2, and the range is y ≥ 0. Their curved shape passes the vertical and horizontal line tests, making them one-to-one functions.

Graphing Strategy: Piecewise functions like f(x) = {x-2 if x<0, 1 if 0≤x≤2, 2x if x>2} require graphing each piece separately within its specified domain. Pay special attention to the transition points to ensure your graph is accurate!

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Algebra 2

•

Jan 3, 2026

•

6 pages

Understanding Functions: Domains, Ranges, and Graphs

Sam

@sam.burchard

Graphing Strategy: Piecewise functions like f(x) = {x-2 if x<0, 1 if 0≤x≤2, 2x if x>2} require graphing each piece separately within its specified domain. Pay special attention to the transition points to ensure your graph is accurate!