Functions I

Contents

Definition

Relation

Any set of ordered pairs (x, y) that connects elements from a first set (domain) to a second set (range). A single input (x) can link to one, zero, or multiple outputs (y).

Function, Expression and Equation

A special type of relation with a strict rule. Every single input (x) has at most one unique output (y). Different inputs are allowed to share the same output, but an input cannot split into two or more outputs.

Term	Definition	Example
Function	A function is a relation (equal sign) between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.	f(x) = 4x + 9
Expression	An algebraic expression is a mathematical phrase (no equal sign) that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply, and divide).	7x² + 2x + 3 42
Equation	A mathematical statement that shows that two mathematical expressions are equal.	For instance, 3x + 5 = 14 is an equation where 3x + 5 and 14 are two expressions separated by an ‘equal’ sign. y = 3x + 5

Difference Between a Function, an Expression, and an Equation

Difference Between a Relation and a Function

Every function is a relation. Because a function is just a relation that obeys an extra rule, all functions automatically qualify as relations.

Not every relation is a function. If a relation allows a single input (x) to point to more than one output (y), it is a relation, not a function.

Relation that is NOT a function:

R_1 = {(1,2), (1,3), (4,5)}

(Here, the input 1 points to two different outputs, 2 and 3, so this is only a relation).
Relation that IS a function:

R_2 = {(1,2), (3,2), (4,5)}

(Here, every input 1 ,3, 4 points to only one output, so it is both a valid relation and a function).

Feature	Relation	Function
Definition	Any collection of ordered pairs	A relation where each input has a unique output
Input Rule (x)	One x-value can map to many y-values	One x-value maps to strictly one y-value
Visual Test	Can fail the vertical line test	Must pass the vertical line test

References

“Algebra – The Definition Of A Function “. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/classes/alg/functiondefn.aspx.

Bhanzu Team. “Relations and Functions — Difference, Types, Examples.” Bhanzu, June 10, 2026. https://bhanzu.com/math/algebra/relations-and-functions.

“Difference Between Relation and Function: Simple Guide.” Vedantu, n.d. Accessed October 17, 2026. https://www.vedantu.com/jee-main/maths-difference-between-relation-and-function.

“Relations and Functions – Definition, Difference, Types, Examples.” CUEMATH. Accessed October 17, 2026. https://www.cuemath.com/algebra/relations-and-functions/.

“What Is A Function? Definition, Types And Notation”. 2022. BYJUS. https://byjus.com/maths/what-is-a-function/.

“What Is the Difference Between Relations and Functions?” Vedantu, December 28, 2023. https://www.vedantu.com/jee-main/maths-difference-between-relations-and-functions.

Vertical Line Test, Horizontal Line Test, One-to-One Function

Vertical Line Test

If no two different points in a graph have the same first coordinate, this means that vertical lines cross the graph at most once. This is known as the vertical line test. Graphs that pass the vertical line test are graphs of functions.

Horizontal Line Test

If no two different points in a graph have the same second coordinate, this means that horizontal lines cross the graph at most once. This is known as the horizontal line test. Functions whose graphs pass the horizontal line test are called one-to-one.

The horizontal line test is a visual method used to determine whether a function is one-to-one (injective) and whether its inverse function is also a function.

One-to-One Function

Graphs that pass both the vertical line and horizontal line tests are one-to-one functions. These are exactly those functions whose inverse relation is also a function. One-to-one functions have an inverse.

In the Fig (a) (which is one to one), x is the domain and f(x) is the codomain, likewise in Fig (b) (which is not one to one), x is a domain and g(x) is a codomain. – CUEMATH

One to one function is a special function that maps every element of the range to exactly one element of its domain i.e., the outputs never repeat.

*Vertical and Horizontal Lines Tests of Functions – Hardy Calculus*

References

“Domain and Range – From Graph.” CUEMATH. Accessed October 17, 2026. https://www.cuemath.com/calculus/domain-and-range-of-a-function/.

“One to One Function – Graph, Examples, Definition.” CUEMATH. Accessed October 17, 2026. https://www.cuemath.com/algebra/one-to-one-function/.

“Vertical Line Test – Definition, Uses, Examples.” CUEMATH. Accessed October 17, 2026. https://www.cuemath.com/algebra/vertical-line-test/.

Krams, Liz. “The Vertical Line Test.” CK-12 Foundation, October 1, 2026. https://flexbooks.ck12.org/cbook/ck-12-elementary-intermediate-college-algebra/section/5.3/primary/lesson/the-vertical-line-test-c-alg/.

Inverse of Functions

A function and its inverse function can be described as the “DO” and the “UNDO” functions. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse function takes the output answer, performs some operation on it, and arrives back at the original function’s starting value. This “DO” and “UNDO” process can be stated as a composition of functions.

A function composed with its inverse function yields the original starting value. Think of them as “undoing” one another and leaving you right where you started.

If functions f and g are inverse functions, f(g(x)) = g(f(x)) = x.

Basically speaking, the process of finding an inverse is simply the swapping of the x and y coordinates. This newly formed inverse will be a relation, but may not necessarily be a function. The original function must be a one-to-one function to guarantee that its inverse will also be a function.

DEFINITION: An inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. If the graph of a function contains a point (a, b), then the graph of the inverse relation of this function contains the point (b, a). Should the inverse relation of a function f(x) also be a function, this inverse function is denoted by f ^-1(x).

If we reflect this graph over the line y = x, the point (1, 0) reflects to (0, 1) and the point (4, 2) reflects to (2, 4). Sketching the inverse on the same axes as the original graph gives us the result in Figure 10.

*Figure 10. The function and its inverse, showing reflection about the identity line – Course Hero*

Note

If the original function is a one-to-one function, the inverse will be a function.

References

⭐ Roberts, Donna. 2022. “Inverse Of Functions – MathBitsNotebook (A2 – CCSS Math)”. mathbitsnotebook.com. https://mathbitsnotebook.com/Algebra2/Functions/FNInverseFunctions.html.

“Use the graph of a function to graph its inverse”. 2022. Course Hero. https://www.coursehero.com/study-guides/collegealgebra1/use-the-graph-of-a-function-to-graph-its-inverse/.

Transformations of Function f (x)

Transformation of functions means that the curve representing the graph either “moves to left/right/up/down” or “it expands or compresses” or “it reflects”. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.

Desmos Examples

Horizontal Right & Left

Vertical Up & Down

Reflection x-axis & y-axis

Vertical Stretch & Shrink

Horizontal Stretch & Shrink

References

“Function Transformations”. 2022. mathsisfun.com. https://www.mathsisfun.com/sets/function-transformations.html.

“Starting with Parent Functions.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGParents.html.

“Functions: Reflections.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGFunctionReflect.html.

“Functions: Vertical Shift.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGFunctionTransVertical.html.

“Functions: Horizontal Shift.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGFunctionTransHorizontal.html.

“Functions: Horizontal Shift.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGFunctionVerticalStretchCompress.html.

“Functions: Horizontal Shift.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGFunctionHorizontalStretchCompress.html.

“Transformations of Functions.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTransformationFunctions.html.

“Functions Transformations – Graphing, Rules, Tricks”. 2022. CUEMATH. https://www.cuemath.com/calculus/transformation-of-functions/.

“Transformations Of Functions | College Algebra”. 2022. courses.lumenlearning.com. https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-transformations-of-functions/.

The Average Rate of Change of a Function

Given a function that models a certain phenomenon, it’s natural to ask such questions as “how is the function changing on a given interval” or “on which interval is the function changing more rapidly?” The concept of average rate of change enables us to make these questions more mathematically precise. Initially, we will focus on the average rate of change of an object moving along a straight-line path.

For a function s that tells the location of a moving object along a straight path at time t, we define the average rate of change of s on the interval [a, b] to be the quantity

Note particularly that the average rate of change of s on [a, b] is measuring the change in position divided by the change in time.

The value of the average rate of change tells us how much the function rises or falls, on average, for each additional unit we move to the right on the graph. For instance, if AV_[3,7] = 0.75, this means that for an additional 1-unit increase in the value of x on the interval [3,7], the function increases, on average, by 0.75 units. In applied settings, the units of AV_[3,7] are “units of output per unit of input”.

The value of the average rate of change is also the slope of the line that passes through the points (a, f(a)) and (b, f(b)) on the graph of f, as shown in the figure below.

*The average rate of change of an abstract function f on the interval [a, b]. – Active Calculus*

Why It Matters

Average rate of change appears throughout algebra, precalculus, and calculus. In algebra, it connects directly to the slope formula you already know. In calculus, it is the starting point for understanding derivatives: the instantaneous rate of change is defined as the limit of the average rate of change as the interval width approaches zero. Real-world applications include calculating average speed, average cost per unit, and average growth rates in science and economics.

Example

Graph the equation -4x² +3x – 4.
Using the given x values [-1, 3], calculate the corresponding y values, and then plot the lines.
(-1, -11), (3, -31)
Calculate the slope of the line using the calculated (x, y) pairs.
m = -5
Calculate the value of b using the slope-intercept form and one of the calculated (x, y) pairs.

y = mx + b

-11 = -5(-1) + b
b = -11 – (-5)(-1) = -11 – 5 = -16

Therefore, the equation of the line is y = –5x – 16
Now plot the line.

*Average rate of change of the function -4x² +3x – 4*

References

⭐ “APC The Average Rate Of Change Of A Function”. 2022. activecalculus.org. https://activecalculus.org/prelude/sec-changing-aroc.html.

“Average Rate of Change.” MathBitsNotebook(A1). Accessed October 17, 2026. https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGAverageRAteChange.html.

“Average Rate of Change Formula: [F(b)−f(a)]/(B−a).” Mathwords. Accessed October 17, 2026. https://www.mathwords.com/a/average_rate_change.htm.

Skelton, Katherine. “Page 4.4: Average Rate of Change.” LibreTexts, July 7, 2023. https://math.libretexts.org/Courses/Queens_College/Preparing_for_Calculus_Bootcamp_(Gangaram)/04%3A_Day_4/4.04%3A_Average_Rate_of_Change.

Positive and Negative Intervals

On a Line

In mathematics, an interval is a set of real numbers between two endpoints. Intervals can be classified as either positive or negative depending on the sign of the numbers in the interval.

If both endpoints of the interval are positive, then the interval is positive. If both endpoints are negative, then the interval is negative. If one endpoint is positive and the other is negative, then the interval contains both positive and negative numbers.

In other words, if we have an interval of the form (a, b), where a and b are real numbers, we can determine its sign as follows:

If a > 0 and b > 0, then the interval is positive.
If a < 0 and b < 0, then the interval is negative.
If a < 0 and b > 0, then the interval contains both positive and negative numbers.

For example, the interval (-3, 5) contains both negative and positive numbers so it is neither a positive nor a negative interval. On the other hand, the interval (-1, -0.5) contains only negative numbers, so it is a negative interval. Similarly, the interval (2, 8) contains only positive numbers, so it is a positive interval.

Of a Function

A function is considered positive when its output values are greater than zero for all inputs in a specified domain. In mathematical terms, if f(x) > 0 for all x in a given interval, then the function is positive over that interval (e.g., its graph sits above the x-axis). This property indicates that the function does not cross or touch the x-axis in that domain. Positive functions are often used in various fields, including economics and physics, to represent quantities that cannot be negative, such as profit or distance. A function is negative if all output values are less than zero f(x) < 0, meaning the graph sits below the x-axis.

Key Ideas

Outputs vs. Inputs: The sign depends on the y-value (output), but the interval itself describes the x-values (input) where that sign happens.
Excluding Zero: Points where the function equals zero f(x) = 0 are never included in positive or negative intervals. They are the dividing lines where a graph crosses the x-axis.
Not Increasing/Decreasing: Being positive or negative tells you where the graph is located (high or low), not whether it goes up or down. A graph can be positive and still slope downward.
The sign of a polynomial between any two consecutive zeros is either always positive or always negative. This is because polynomial functions are continuous functions (no breaks in the graph), which means that the only way to change signs is to cross the \[x\]-axis. But if this happened, the given zeros would not be consecutive!

How to Find Them

Look at a Graph: Find the x-ranges where the curve is above the x-axis (positive) or below it (negative). Use parentheses for the interval bounds (e.g., (-3, 3)) to leave out the zeros.
Test a Value: For continuous functions like polynomials, find where f(x) = 0 first. Pick an x-number between those zeros, plug it into the function, and see if the result is plus or minus.

Example

Positive and Negative Intervals – Khan Academy

For example, consider the graphed function f(x) = (x + 1)(x – 1)(x – 3).

From the graph, we see that f(x) is always

negative when ∞ < x < -1.
positive when -1 < x < 1.
negative when 1 <x < 3.
positive when 3 < x < ∞.

It is not necessary, however, for a polynomial function to change signs between zeros.

For example, consider the graphed function g(x) = x(x+2)².

From the graph, we see that g(x) is always

negative when ∞ < x < -2.
negative when -2 < x < 0.
positive when 0 < x < ∞.

Notice that g(x) does not change sign around x = -2.

References

Alexander. “Positive Function / Negative Function.” Statistics How To, March 7, 2022. https://www.statisticshowto.com/positive-function/.

Anonymous. “What Does It Mean for a Function to Be Positive?” Answers, Accessed August 16, 2026. https://math.answers.com/math-and-arithmetic/What_does_it_mean_for_a_function_to_be_positive.

[ ℰ ] “Features of Function Graphs.” MathBitsNotebook, Accessed August 16, 2026. https://www.mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGFunctionFeatures.html.

“Increasing, Decreasing, Positive or Negative Intervals (Video).” Khan Academy, June 25, 2015. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:intervals-where-a-function-is-positive-negative-increasing-or-decreasing/v/increasing-decreasing-positive-and-negative-intervals.

[ ℰ ] “Lesson 2: Positive and negative intervals of polynomials” Khan Academy, Accessed August 16, 2026. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-intervals/a/positive-and-negative-intervals-of-polynomials.

“Positive and Negative Intervals: Formula, Solved Examples & FAQs.” eTutorWorld, Accessed August 16, 2026. https://www.etutorworld.com/math/positive-and-negative-intervals.html.

“Positive and Negative Intervals of Polynomials (Video).” Khan Academy, July 20, 2019. https://www.khanacademy.org/math/get-ready-for-ap-calc/xa350bf684c056c5c:get-ready-for-applying-derivatives-to-analyze-functions/xa350bf684c056c5c:positive-and-negative-intervals-of-polynomials/v/polynomial-intervals.

“What Are Graphs And Behavior Of Functions?” UPI Study, June 28, 2026. https://www.upistudy.com/blog/calculus-1/what-are-graphs-and-behavior-of-functions.

Increasing, Decreasing and Constant Functions

In mathematics, as we know that a function is a relation between input and output. A function can be increasing, decreasing, or constant for the given intervals throughout their entire domain, and they are continuous and differentiable in the given interval. Now, what is an interval: so, an interval is known as a continuous or connected part or portion on the real line. This increasing or decreasing of function is generally used in the application of derivatives. So, if you want to find that the given function is increasing or decreasing in the given interval then you can easily find it with the help of derivatives.

Increasing

When a function is increasing in the given interval, then such type of function is known as increasing function. Or in other words, when a function, f(x), is increasing, the values of f(x) are increasing as x increases. Or, let us considered I be an interval which presents in the domain of a real valued function f. Then the function f is increasing on I, if x1 < x2 in I ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ I. Or, in terms of derivative, a function is increasing when the derivative at that point is positive. The graphical representation of an increasing function is:

Decreasing

When a function is decreasing in the given interval, then such type of function is known as decreasing function. Or in other words, when a function, f(x), is decreasing, the values of f(x) are decreasing as x increases. Or, let us considered I be an interval which presents in the domain of a real valued function f. Then

The function f is decreasing on I, if x1, x2 in I ⇒ f(x1) < f(x2) ∀ x1, x2 ∈ I.
The function f is decreasing on I, if x1 < x2 in I ⇒ f (x1) ≥ f(x2)∀ x1, x2 ∈ I.
The function f is strictly decreasing on I, if x1 < x2 in I ⇒ f(x1) > f(x2)∀ x1, x2 ∈ I.

Or, in terms of derivative, a function is decreasing when the derivative at that point is negative. The graphical representation of a decreasing function is:

Constant

When a function is neither increasing nor decreasing in the given interval, then such type of function is known as constant function. Or in other words, when a function, f(x), is constant, the value of f(x) does not change as x increases. Or, let us considered I be an interval which presents in the domain of a real valued function f. Then the function f is constant on I, if f(x) = c ∀ x ∈ I. Here, c is a constant. Or, in terms of derivative, a function is constant (i.e. neither increasing nor decreasing) when the derivative is zero. The graphical representation of constant function is:

References

“Increasing And Decreasing Functions – GeeksForGeeks”. 2021. GeeksForGeeks. https://www.geeksforgeeks.org/increasing-and-decreasing-functions/.

“3.3: Increasing And Decreasing Functions”. 2017. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Apex)/03%3A_The_Graphical_Behavior_of_Functions/3.03%3A_Increasing_and_Decreasing_Functions.

“How to Find the Increasing or Decreasing Functions?” 2022. EffortlessMath. https://www.effortlessmath.com/math-topics/how-to-find-the-increasing-or-decreasing-functions/.

“Increasing And Decreasing Functions – Definition, Rules, Examples”. 2022. CUEMATH. https://www.cuemath.com/calculus/increasing-and-decreasing-functions/.

“Increasing And Decreasing Functions”. 2022. mathsisfun.com. https://www.mathsisfun.com/sets/functions-increasing.html.

“Increasing/Decreasing Functions”. 2022. CliffsNotes. https://www.cliffsnotes.com/study-guides/calculus/calculus/applications-of-the-derivative/increasing-decreasing-functions.

Function: Continuous or Discontinuous (Infinite, Jump or Removeable)

In calculus, a function is continuous at x = a if and only if all three of the following conditions are met:

The function is defined at x = a; that is, f(a) equals a real number
The limit of the function as x approaches a exists
The limit of the function as x approaches a is equal to the function value at x = a

There are three basic types of discontinuities:

Removable (point) discontinuity – the graph has a hole at a single x-value. Imagine you’re walking down the road, and someone has removed a manhole cover (Careful! Don’t fall in!). This function will satisfy condition #2 (limit exists) but fail condition #3 (limit does not equal function value).
Infinite discontinuity – the function goes toward positive or negative infinity. Imagine a road getting closer and closer to a river with no bridge to the other side
Jump discontinuity – the graph jumps from one place to another. Imagine a superhero going for a walk: he reaches a dead end and, because he can, flies to another road.

Both infinite and jump discontinuities fail condition #2 (limit does not exist), but how they fail is different. Recall for a limit to exist, the left and right limits must exist (be finite) and be equal. Infinite discontinuities have infinite left and right limits. Jump discontinuities have finite left and right limits that are not equal.

Examples

Example 1

Is f(x) continuous at x = 0?

To check for continuity at x = 0, we check the three conditions:

Is the function defined at x = 0? Yes, f(0) = 2
Does the limit of the function as x approaches 0 exist? Yes
Does the limit of the function as x approaches 0 equal the function value at x = 0? Yes

Since all three conditions are met, f(x) is continuous at x = 0.

Example 2

Is f(x) continuous at x = -4?

To check for continuity at x = -4, we check the same three conditions:

The function is defined; f(-4) = 2
The limit exists
The function value does not equal the limit; point discontinuity at x = 4

Since all three conditions are NOT met, f(x) is not continuous at x = 4.

Example 3

*Section 2.9 : Continuity – Paul’s Online Notes*

Example 4

References

“Continuity in Calculus: Definition, Examples & Problems”. 2022. study.com. https://study.com/academy/lesson/continuity-in-calculus-definition-examples-problems.html.

“Calculus I – Continuity”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx.

Additional Reading

SepiaGlyphs. “How Did Mathematicians Historically Develop the Modern Definition of Functions?” Quora, October 19, 2026. https://qr.ae/pFlC8S.

The concept of a mathematical function evolved significantly over centuries. Initially viewed by Leibniz as a geometric property, Euler later redefined it as an algebraic formula. However, complexities like the plucked violin string and Joseph Fourier’s research on infinite series challenged this definition. Ultimately, Peter Gustav Lejeune Dirichlet moved beyond formulas, defining a function as any arbitrary correspondence between inputs and outputs, eventually formalized through modern set theory as ordered pairs.

[ ℰ ] This exceptional reference is highly recommended for your consideration.

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The featured image on this page is from the page “What is a function?” 2022. thinglink.