Functions I

Definition

A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.

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, in which 3x + 5 and 14 are two expressions separated by an ‘equal’ sign.

References

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

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

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.

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.

References

“Vertical and Horizontal Line Tests”. 2022. hardycalculus.com. http://hardycalculus.com/calcindex/IE_verticalline.htm.

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.

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.

References

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

“Functions Transformations – Graphing, Rules, Tricks”. 2022. CUE MATH. 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.

Example

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

References

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

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:

1. The function is defined at x = a; that is, f(a) equals a real number
2. The limit of the function as x approaches a exists
3. 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:

1. 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).
2. 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
3. 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:

1. Is the function defined at x = 0? Yes, f(0) = 2
2. Does the limit of the function as x approaches 0 exist? Yes
3. 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:

1. The function is defined; f(-4) = 2
2. The limit exists
3. 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

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.

---

The featured image on this page is from the page “What is a function?” 2022. thinglink. http://www.thinglink.com/scene/538088438402908160.

---

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.