# High School Functions Worksheets

**Why do We Study Math Functions?** - We know what the term "function" generally means. But in math, a function is a relation in which no input relates to more than one output. This means that a function is a relation with one output for each input.
These functions are the bases of all mathematical operations. With functions, we learn how to solve various algebraic equations and it is the base platform for calculus. You will learn how to place them on the graph and solve equations through them. Functions teach us a lot of techniques to solve problems, alongside developing the skills to make sense of various problems and helping us to find the relevant, appropriate method in solving them. Functions also help in constructing viable arguments to support your answers. They also help in increasing concentration and help you in learning how to structure the questions that need to be solved. Functions are a very elusive concept for many students. They help us understand the world around us and are essential in the business world. Have you ever thought of buying a car or calculated how long it will take you to get to a location (while accounting for other variables); then you have come across functions before.

### Interpreting Functions

- Relations as Functions (HSF-IF.A.1) - Relations are sets that have inputs and outputs. A function is a relation that has a single output per input given.
- Domains and Ranges of Functions (HSF-IF.A.2) - Domain is a cumulative term for all the possible inputs a function has. The range is the possibility of what can come from those inputs.
- Evaluating Functions (HSF-IF.A.2) - These worksheets show students how to get comfortable with working with functions.
- Evaluating Advanced Functions (HSF-IF.A.2) - These can be a little more clunky, at first, for students.
- Variable Expressions and Sequences (HSF-IF.A.3) - Understanding all the aspects of this topic can be a bit overwhelming.
- Functions versus Relations (Solutions Included) (HSF-IF.B.5) - Students will be able to separate the two concepts here.
- Determining and Predicting the Rate of Change of Functions (HSF-IF.B.6) - This is a skill that directly applies to many forms of science.
- Graphing Linear and Quadratic Functions (HSF-IF.C.7a) - Being able to place these functions on a graph is a fundamental skill. You can often get away with plotting points of your own.
- Graphing Square and Cube Roots (HSF-IF.C.7b) - There is a set trend that follows each form.
- Graphing Polynomial Functions (HSF-IF.C.7c) - Start with the intercepts and then predict the symmetry that may exist.
- Graphing Rational Functions (HSF-IF.C.7d) - This is a more primitive version than you will see in advanced classes.
- Graphing Exponential and Logarithmic Functions (HSF-IF.C.7e) - We not only plot them but start to use the graphs to make predictions based on the lines.
- Classifying Even and Odd Functions (HSF-IF.C.8) - It all comes don to where they fall on the graph and symmetry.
- Expressions for Exponential Functions (HSF-IF.C.8b) - We help this topic take on other forms.
- Comparing Functions in Different Formats (HSF-IF.C.9) - We use the different formats because they make it more understandable for our audience. We show you the uses of each of them.

### Building Functions

- Explicit Expressions and Recursive Processes (HSF-BF.A.1a) - Students will learn how to reshape formulas and expressions. These topics have a good deal of application computer programming.
- Exponential Decay (HSF-BF.A.1b) - We use the rate of decay to determine the end value of many practical situations.
- Composition of Functions (HSF-BF.A.1c) - We understand the concept top to bottom after this.
- Manipulating the Graphs of Functions (HSF-BF.B.3) - Learn how to use a graph to your advantage.
- Inverses of Discrete Functions (HSF-BF.B.4a) - We learn how to put them in reverse and make use of the outcome.
- Graphing The Inverse of Functions (HSF-BF.B.4c) - These can help us pull apart a function and make it more useful based on what we are faced with. You will learn the math and applications behind this.
- The Inverse Relationship of Logarithms and Exponents (HSF-BF.B.5) - These polar opposites can be used to help us better understand a situation.
- Invertible Functions (HSF-BF.B.4d) - This is when a one to one relationship exists between inputs and outputs.
- Converting Between Logarithmic and Exponential Functions (HSF-BF.B.5) - These are more interchangeable then you will first realize. You will understand how to use both forms and convert between them.

### Linear, Quadratic, & Exponential Models

- Comparing Linear and Exponential Functions (HSF-LE.A.1a) - I remember this as linear (meaning line) are straight lines and exponentials are curves. That is when you throw them up on a graph.
- Constructing Linear, Quadratic, and Exponential Models of Data (HSF-LE.A.2) - This is not much of a curve ball for you at all. This is what I would call practical math. You will use this in the real world.

### Trigonometric Functions

- Radians, Degrees, and Arc Length (HSF-TF.A.1) - These are all fundamental measures of circle math.
- Using and Understanding the Unit Circle (HSF-TF.A.2) - It is important to note that the circle is placed at the origin (0,0) when this is examined.
- Using the Unit Circle Reference Angles (HSF-TF.A.2) - These distinct divisions of the unit circle help us make accurate predictions. Why not work through this work on known angles.
- Using the Unit Circle with Trigonometric Identities (HSF-TF.A.3) - This is when you apply right triangles to the unit circle.
- Symmetry of the Unit Circle and Odd-Even Properties (HSF-TF.A.4) - Each form of behavior is explained and examined here. It will have a great deal of application for you as you advance.
- Modeling Phenomena with Trigonometric Functions (HSF-TF.B.5) - This has huge applications in a variety of engineering calculations. This is a good way to first start exploring how to use this.
- Applying Trigonometric Identities (HSF-TF.B.5) - There is a series of rules and common occurrences that can help us learn more about a system.
- SohCahToa (HSF-TF.B.5) - You will learn how to put this to work for you.
- Modeling Periodic Phenomena with Trigonometric Functions (HSF-TF.B.6) - When a phenomenon occurs at a predictable interval, we have a great range of applications for it.
- Pythagorean Trigonometric Identities (HSF-TF.C.8) - As usual, right triangles make the world go around in geometry.

### What is the Importance of Understanding Functions?

In real life we often talk or define relationships between people. For example, there may be a boy in the street that you waive to and a friend may ask you how you know him. That boy could be a friend, a cousin, a brother, a teammate, and maybe you were just being courteous to a stranger. When you define that relationship to your friend, you have basically stated how important that person is to you. In a way you have defined an unspoken value for that person in relation to you. Functions are mathematical methods of describing relationships between values and salient variables. These relationships often require complex computations. This is why you often will not start to explore the higher performing functions until you enter undergraduate school. Students that master the ability to not only interpret well stated functions but write and create their own possess a skill that will be in high demand in the corporate world. This is because if you master this skill, you will quickly be able to judge the significance of trends and spot outliers within the data. People that understand the essence of the nature of a function can often make exactly accurate predictions of how data flows through it. This allows them to make solid decisions based on what the data shows them. Functions find themselves used for all types of modeling whether it financial, an engineering application, or trajectory of a space shuttle as it approaches Mars. The people that can understand and create these models with often secure top level employment quickly.

### How Are Math Functions Used in Real World?

The funny thing is that there are almost too many applications of functions in the real world to list or at least do justice for. Every time you make a move, at all, that can be broken down mathematically that results in some kind of outcome (output) you have taken part in a mathematical function. Let me just recount how many functions I came across just last night:

1. I put 75 cents in coins into a vending machine and then chose the letters that indicated a fruit roll. The machine in turn moved a coil that the fruit roll was on, and the fruit roll popped out.

2. I received my paycheck which I am paid hourly for. Depending on the number of hours that I work, I receive a different amount each time.

3. I took an Uber home. The cost of that ride is based on the distance covered and amount of time involved.

4. I used my television remote control to tune in the basketball game. Do you see where I’m going with this?