Helpful Tools for Calculus, Chapter 1: Functions and Relations
Functions
Let’s begin with a very basic definition: a function is a relationship between two variables. To be more specific we could say that there is an input and an output and the function defines how to get from one to the other. Generally a function has a name, like f or g, or a description, like “the amount of potatoes sold over time”, that helps to define and describe it. The names that we attribute to a function are often related to what it describes, for example “the amount of potatoes sold over time” might be expressed as P(t) where P is the name of the function that stands for potatoes and the (t) part tells us that the variable t, representing time in this case, is what is being ‘plugged in’ to this function.
Note that in this case the parenthesis are NOT representing multiplication. Multiplication between two variables should never be expressed with parenthesis because of the notation here that would define it as a function.
In functions, we generally have one or more inputs and one output. The input could be referred to as what we ‘plug in’ and the output is what we ‘get out’ of the function. The list of possible inputs is called the domain and is denoted D. The list of possible outputs is called the range and is denoted R. Again, be careful. We mentioned that the set of real numbers is represented with 
, so make sure to distinguish these two when writing. Generally, we use x to represents something that gets plugged in and y to represent the output. So x∈D and y∈R.
Now let’s clean up our definition a little bit. Technically a relation is a relationship between two variables. A function is a specific type of relation where there is a only one y-value (output) for each x-value (input).
More formally:
Given two sets A and B, a function from A to B is a mapping that takes each element x∈A and relates with it a single element y∈B. We write f:A→B, where f is the name of this function. Given an element of A (an x-value), f(x) is used to represent the element of B (a y-value) that is related to x through f. A is called the domain of f while a similar thing cannot be said about B. We can say that the range of f is a subset of B so that {y∈B: y=f(x) for some x∈A}. This is read as “y is an element of B if y equals f(x) for some x value in the domain.” We say that the domain of a function exists when x is between certain numbers: a < x < b. Similarly, for y: c< y< d. We often use a bracket/parenthesis notation. For example: x∈ (a,b) and y ∈ (c,d).
For the function f(x)= x2+1 we say the domain is all real numbers, or x ∈ and the range is 1 and above so y ∈ ≥ 1 or just y ≥ 1 since we generally assume that we are talking about real numbers. Other ways to say this are: x ∈ (-∞,∞) and y∈ [1,∞).
Graphing
One of the best ways to visualize a function is with a graph. A graph is generally represented on a Cartesian Plane named after René Descartes, the mathematician who created it. A Cartesian Plane is just a two-dimensional space, up-down and left-right, used to represent the relationship between two variables. The dependent variable, usually x or t (time), is plotted in the horizontal direction while the independent variable, usually y or f, is plotted in the vertical direction. Since both directions can represent positive and negative values, there are four quadrants in addition to the two axis:
The quadrants are organized counter-clockwise starting in the to right. So the section where everything is positive is the first quadrant and the section where everything is negative is the third quadrant. These labels are not normally placed on the graph. The origin is the location at the center, where the coordinates at (0,0). However, it is often important to label the axis with the variable that it represents. It is very important to make sure the graph is scaled consistently. That is to say, each notch on the axis is equally spaced out if they represent the same values. (There are types of graphs that don’t do this. Those are generally used for specific purposes, like logarithmic scaling). The center of the graph, where the axis intersect is called the origin. It is OK to change the scaling and the focus of a graph depending on the situation, so the origin or axis may not even be visible.
In order to graph an equation, it is helpful to calculate the coordinate pairs by picking some values from the domain to plug into the function. If you are unfamiliar with the function, it is a good bet to pick some whole numbers between -10 and 10 to plug in for x. This can be organized in an xy-chart like the following:
This tells us that some of the points on the graph are (-5,7), (-2,4), (0,2), (2,0) and (5,-3). Next, we plot these on a graph and then connect them with a line as best as we can:
Example 1: Graph y=1.5x+1
Solution:
Example 2: Graph the function f(x)=x2+2x-3
Solution:
Monotonicity and the Vertical Line Test
Let’s talk about the directions that a line on a graph can go.
For a relationship between x and y to be a function, remember there has to be only one output for each input. That is to say, there cannot be more than one y-value for each x-value. Identifying if a graph is representative of a function is often done by using The Vertical Line Test. The test is simply to see that, given a vertical line, if there is any portion of the graph that crosses the vertical line more than once. If so, then it is not a function. An example of a relation which fails the vertical line test looks like this:
Calculus and Algebra begin with functions that pass the vertical line test because they are often easier to evaluate and study. In particular, a topic that is important is identifying whether or not a function is increasing or decreasing. In a graph like the one above, it is difficult to define and clarification is needed for the region around the loop.
If a function is strictly increasing, like y=x, we call it monotonic increasing. Monotonicity in a function means that the function is only going that direction. So y=-x is monotonic decreasing, while y=x2 is not monotonic because it both decreases and increases. This description because very important later on when discussing limits.
Mathematically, we say a function f(x) is monotonic increasing if f(a)≤f(b) where a,b∈D and a<b. A function is strictly increasing if f(a)<f(b). The definition for monotonic and strictly decreasing are very similar.
Roots
A root is an x-intercept of a function. This is where the function will have a y-coordinate of 0. To solve for a root, set the function equal to 0. For polynomials, this requires factoring. For other functions it will be more complicated. Generally, use the reverse order of operations to isolate your variable. The roots are sometimes called zeroes, solutions, and x-intercepts, depending on the context.
Translation (Shifting)
Vertical translation is when you add one specific number, or constant, to a function. For example, f(x) = x2 + 3 is translated vertically by 3 units. If we subtract a number, then we are moving the function down instead of up.
Horizontal translation is when you add one specific number, or constant, to the input of a function. So, f(x) = (x+3)2 is moved to the left 3 units. This one makes the graph move opposite to what number is put inside it. So, f(x) = (x-5)2 is moved right 5 units.
Translation is generally of the form y=f(x-a)+b where a is the horizontal movement and b is the vertical movement.
Scaling (Stretching)
Vertical scaling occurs when multiplying a function by a constant. If the number is positive, then the function retains its normal shape. If the number is negative, then the function flips upside down. If the number is bigger than 1 then the function gets taller, or stretches vertically. If the number is smaller than 1 then the function gets shorter, or shrinks.
Horizontal scaling occurs when multiplying the input of a function by a constant. Multiplying by a negative number flips the graph horizontally. Multiplying by a number larger than 1 makes the graph ‘happen’ faster, while multiplying by a fraction makes the graph ‘happen’ slower. Imagine grabbing the ends and either pulling or squishing.
Scaling is generally of the form y=b∙f(ax) where a is the horizontal shrinking and b is the vertical stretching.
Combining Functions
It is possible to combine two functions using operations. The sum of two functions f and g is simply the two functions added together with all domain restrictions of either function. The product of two functions is the two functions multiplied together with all domain restrictions of either function. The quotient of two functions f and g is the result of dividing f by g with the added domain restriction(s) of g≠0. Finally, the composition of two function is created by plugging one function into the other and is denoted with a small, empty circle, ◦, or by using parenthesis to note the relationship, f(g(x)).
Domain
To find the domain of a function there are a couple of key things we can look for. We can start by assuming that all functions have a domain consisting of all real numbers before looking for things that may restrict the domain. The main things that can cause domain restrictions are roots and division. This is because we cannot solve roots of negative numbers when working in the real numbers and we cannot divide by 0. There are a few other things that cause domain restrictions which will be mentioned as they come up.
To calculate the domain, we can purposefully look for these things that cause a domain restriction, find out when they happen, and then clarify that the domain does not include those values.
Example 3: Identify the domain of y=2x+7.
Solution: There are no domain restrictions, so the domain is x∈ or x∈(-∞,∞).
Example 4: Identify the domain of 
Solution: Since there is a square root, we need to ensure that it cannot be negative. So we say:
Thus, the domain is x≥-3 or x∈[-3,∞).
Example 5: Identify the domain of 
Solution: Since there is a fraction, we need to ensure that the denominator cannot be 0. We say:
Thus, the domain is 
It is not necessary to write the domain in multiple different ways. It is acceptable to choose the notation that makes the most sense for the situation and is most concise.
Range
To identify the range of a function it is helpful to plug in some values of the domain into the function and check what happens at extremes. For the most part, one should look for trends in the function and focus less on specific values. For example, y=3x has no domain restrictions so you can plug in anything you want. If we choose some values for x, we can see the following:
It seems that the y-value can be a large negative number, a large positive number, and everything between. So we might say that the range of y=3x is, like the domain, also all real numbers, or y∈ . Putting that together, if f(x)=3x then f: →.
Example 6: Identify the domain and range of f(x)=2x2+1
Solution:
Domain: There are no square roots or fractions, so the domain is x∈.
Range: When plugging in values to the function we see that large positive numbers appear, but no large negative numbers appear. Upon making an xy-chart and a graph, we can see that the smallest value of the function is y=1. So, the range can be written as y≥1 or y∈[1,∞).
Conclusion: f: →[1,∞)
Another method of identifying the range is to learn the parent functions.
Parent Functions
Parent functions are functions that describe a specific group of functions with the same characteristics. The parent functions are generally thought of as the following:
Polynomial: made up of the sum of a single variable raised to non-negative exponents.
3x5-5x4+2x2+x+15
Rational: a ratio of two polynomials.
Algebraic: Made up of sums, products, quotients and roots of rational functions.
Exponential and Logarithmic: Made up of terms raised to a variable or defined by the logarithm of a variable.
2x or log(3x-2)
Trigonometric: Built from sine and cosine functions (cosecant, secant, cotangent, tangent).
sin(x) or 3cot(x)
We know that all polynomials have a domain and range consisting of the real numbers. Rational functions require calculating the zeroes of the denominator, and therefore will usually have domain restrictions and possibly asymptotes. Algebraic functions will vary based on the exponent. Odd Algebraic functions have no domain or range restrictions often, while even algebraic functions have a domain restriction so that the input of the function must not be negative. Exponential functions have no domain restrictions, but the range is y>0 while logarithmic functions are the inverse with a domain of x>0 and range y∈. Trigonometric Functions have a domain restriction if it is defined by the quotient of other functions, like tangent, and the domain is either -1≤y≤1, y≠0, or y∈ depending on the specific function. It could be very helpful to take the time to graph some of these functions and identify their general shapes and domains. Also, on the Algebra and Trigonomotry Cheat Sheet, there are graphs of each of these parent functions and more.
The Distance between Points
When comparing parts of a function, knowing the distance between some two points could be helpful in determining important information. The distance between two points can easily be deciphered by drawing a triangle connecting the two points as show below. By using the Pythagorean Theorem, we can calculate the distance between the two points.
We can label the bottom of the triangle as ∆x and the right side as ∆y because the length of those sides is the change between those respective variables. So, the distance between the points, which is the slanted side, can be solved by doing the following:
Sometimes this is rewritten for clarity so ∆x=x2-x1 and ∆y=y2-y1. Thus,
Example 7: Identify the distance between the two points (3,7) and (1,4).
Solution:
Review
A function is a mapping between two variables, usually x and y.
A domain is the list of possible inputs or x-values for a function.
A range is the list of possible outputs or y-values for a function.
The Vertical Line Test is a way to determine if a graph is representative of a function.
Monotonicity describes if a function is only increasing or only decreasing.
Parent Functions can help to identify domains and ranges of functions quickly.
The root of a function is when the y-value is equal to 0 and can be called the x-intercept.
The y-intercept occurs when the x-value of a function is equal to 0.
The distance between two points on a graph can be determined using the Pythagorean Theorem.