What is a function

Here you will learn about functions in algebra, including what functions are, how to calculate with function machines and exponential functions.

Students will first learn what is a function as part of functions in katex is not definedth grade and continue to learn about them in high school.

What is a function?

A function has the relationship where there is one unique output for each input.

A function can be represented by a…

Let’s look at another example and a non-example of each type.

For example,

Functions in algebra are used to describe the operation being applied to an input in order to get an output.

Function machines

Functions can be presented using a function machine.

Step-by-step guide: Function machine

Function notation

Functions can be described using function notation.

The katex is not defined in katex is not defined is known as the function that is being applied to a variable katex is not defined Other letters such as katex is not defined and katex is not defined are also commonly used.

For example,

Let’s look at a function katex is not defined described by a function machine:

Then let’s rewrite this using function notation:

An exponential function is in the form katex is not defined and is a mathematical function where katex is not defined and katex is not defined are variables, and katex is not defined and katex is not defined are constants, katex is not defined

For example,

The diagram shows the graphs of katex is not defined and katex is not defined

The domain of the functions is all real numbers. Let’s consider the range of the functions. It may appear that the functions reach the katex is not defined-axis on the graph of the functions, but they do not.

They are actually continually getting closer and closer to the katex is not defined-axis, but will never intersect. This is a horizontal asymptote at katex is not defined (the katex is not defined-axis) and it occurs because katex is not defined can never equal zero.

What is a function?

Common Core State Standards

How does this relate to katex is not definedth grade math and high school math?

• Grade 8 – Functions (8.F.A.1)
  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

• Functions – Interpreting Functions (HSF.IF.C.7e)
  Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

• Functions – Linear, Quadratic and Exponential Models (HSF.LE.A.2)
  Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).

How to decide what is a function

In order to decide what is a function:

1. Decide if each inputkatex is not defined has ONE unique outputkatex is not defined.
2. If so, it is a function. If not, it is not a function.

What is a function examples

Example 1: is the graph a function?

Is the graph a function?

1. Decide if each inputkatex is not defined has ONE unique outputkatex is not defined.

All inputs (katex is not defined values) correspond to only one point on the line – one output (katex is not defined value).

2If so, it is a function. If not, it is not a function.

The graph IS a function.

Example 2: is the table a function?

Is the table a function?

How to use function machines

In order to solve an equation using a function machine:

1. Consider the order of operations being applied to the unknown.
2. Draw a function machine starting with the unknown as the Input and the value of the equation as the Output.
3. Work backwards, applying inverse operations to find the unknown Input.

Function machines example

Example 3: solving a one step equation using a function machine

Solve katex is not defined.

Draw a function machine starting with the unknown as the Input and the value the equation is equal to as the Output.

Example 4: solving a two step equation using a function machine

Solve katex is not defined.

Draw a function machine starting with the unknown as the Input and the value of the equation as the Output.

How to find the equation of exponential functions

In order to find the equation of exponential functions:

1. Find two points that lie on the graph.
2. Form two equations in the form katex is not defined
3. Solve the equations simultaneously.

Step-by-step guide: Exponential function

Example 5: finding exponential functions given the y-intercept

Find the equation of the exponential function in the form katex is not defined

Example 6: finding exponential functions by dividing the equations

Find the equation of the exponential function in the form katex is not defined

Example 7: finding exponential functions by dividing the equations

Find the equation of the exponential function in the form katex is not defined

Teaching tips for what is a function

• When first teaching functions, focus on identifying whether or not something is a function and not necessarily identifying the function name (type of function).

• Expose students to more than just linear functions, by including a variety of other types, such as quadratic, exponential or polynomial functions.

• Represent functions and non-functions in a variety of ways. While students may find the vertical line test useful, ensure that this is not the only way they can identify a function.

• Let struggling students use calculators, so the focus is on the function and not the calculations.

Easy mistakes to make

• Thinking only graphs or equations are functions
  While these are common ways to represent functions, it is important to know that functions can also be represented by tables or descriptions.
  For example,
  
  
  Description: The input, katex is not defined cubed is equal to the output, katex is not defined
  The table and description represent a function since each input has one unique output.

• Thinking every function must have an equation
  Many functions can be represented by an equal, but this is not a requirement.
  Some functions cannot be represented by an equation.
  For example,
  
  
  The graph represents a function since each input has one unique output, but no equation fits this function.

• Thinking katex is not defined values can’t repeat for different katex is not defined values
  The domain of a function is the katex is not defined values and they must each have a unique output. However, the range of a function, the katex is not defined values, may repeat.
  
  In the graph of the function above, notice that katex is not defined for both katex is not defined and katex is not defined

• Students who think the input values or output values are always all real numbers
  Sometimes the input and output value are all real numbers, but this is not always the case. Either the domain or range or both can be a subset, such as just integers or whole numbers. This is why it is important to always define the domain and range of a function.

• Not using the correct order of operations when drawing a function machine
  A common error is to not follow the correct order of operations when creating a function machine for an equation.
  
  For example, for the equation katex is not defined the multiplication by two takes place before subtracting one.

• Function notation is mistaken for a product
  It is common for katex is not defined to be thought of as katex is not defined times katex is not defined rather than katex is not defined of katex is not defined This confusion can lead to incorrect evaluations of values.
  
  For example,
  When finding katex is not defined when katex is not defined a student may think katex is not defined means katex is not defined
  
  The correct solution is,
  katex is not defined

• Incorrectly applying the exponent to both variables
  It is easy to assume that katex is not defined with katex is not defined are equal. However, katex is not defined means katex is not defined and this means katex is not defined, which is a different function.

The next lessons are

• Law of exponents
• Scientific notation
• Quadratic graphs

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