# Function Calculator

## Calculate the properties of a function step by step

The calculator will try to find the domain, range, x-intercepts, y-intercepts, derivative, integral, asymptotes, intervals of increase and decrease, critical (stationary) points, extrema (minimum and maximum, local, relative, absolute, and global) points, intervals of concavity, inflection points, limit, Taylor polynomial, and graph of the single-variable function. The interval can be specified. Parity will also be determined.

Enter a function of one variable:
Enter an interval:
Required only for trigonometric functions. For example, [0, 2pi] or (-pi, oo). If you need oo, type inf.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Function Calculator is a tool that allows you to many properties of functions. Easily explore functions by examining their parity, domain, range, intercepts, critical points, intervals of increase/decrease, local and global extrema, concavity intervals, inflection points, derivatives, integrals, asymptotes, and so on.

## How to Use the Function Calculator?

• ### Input

Enter the function you want to analyze. If necessary, enter the interval which you are interested in.

• ### Calculation

Click the "Calculate" button to start the process.

• ### Result

The calculator will quickly display the properties of the function.

## What Is a Function?

In mathematics, a function is a formula that, for every value from the input set (domain), produces some value. The set of such values forms the output set (range, codomain). The function is called one-to-one if no two elements from its domain give the same element from the range.

Functions are typically denoted using various notations. The most common notation is $f(x)$. In the notation $f(x)$ $f$ is the function symbol and $x$ represents the input variable.

For example, $f(x)=x^2$ is a quadratic function. If $x=-2$, then $f(-2)=(-2)^2=4$. If $x=2$, then $f(2)=2^2=4$. By the way, it is not one-to-one, since there are two elements, namely $-2$ and $2$, that give $4$.

## What Are the Components of a Function?

• Domain: The domain of a function is the set of all possible input values. It defines where the function is defined or valid.
• Codomain: The codomain of a function is the set of all possible output values. It represents the values that the function can produce.
• Range: The range of a function is the set of all actual output values. It represents the values that the function actually produces.
• Input: The input of a function is the value of its variable you provide as an argument to the function. It's the value for which you want to determine the corresponding output.
• Output: The output of a function is the result of applying the function to the input value(s). It represents the dependent variable's value corresponding to the given input.

## What Is the Mathematical Expression of a Function?

A function is often expressed using a formula or rule that describes how the input values are transformed into output values. This formula can be very simple, for example, $f(x)=x$, or very complex, for example, $f(x,y,z)=\frac{x^2+y^2z}{z-xy}$ (a multivariable function). Here are some examples:

• Linear Function. A linear function has the following form:

$$f(x)=mx+b$$

Example: $f(x)=2x+3$.

This function represents a straight line on a graph, where $m$ is the slope, and $b$ is the y-intercept.

$$f(x)=ax^2+bx+c$$

Example: $f(x)=3x^2-4x+1$.

This function represents a parabolic curve.

• Exponential Function. An exponential function has the following form:

$$f(x)=a\cdot b^x$$

Example: $f(x)=2\cdot3^x$.

This function exhibits exponential growth or decay.

## Why Choose Our Function Calculator?

• ### Ease of Use

Our calculator is designed with user-friendliness in mind. It features an intuitive interface that allows both beginners and experts to navigate with ease.

• ### Versatility

Our calculator can handle a wide range of functions to meet your requirements.

• ### Accurate Results

Our calculator provides accurate results. You can trust it.

### FAQ

#### What is meant by the graph of a function?

The graph of a function is a visual representation that shows how the output (dependent variable) of the function relates to the input (independent variable). It is a plot of points where each point corresponds to an input-output pair of values. The graph illustrates the behavior of a function—its shape, intersection points, critical points, inflection points, and so on.

#### What is the difference between a relation and a function?

A relation is a set of ordered pairs $(x,y)$, where $x$ is the input (independent variable), and $y$ is the output (dependent variable). It can include multiple y-values for the same x-value. In contrast, a function is a special type of relation, which is defined by some rule, so you can calculate the output value, given the input value.

#### What is the Function Calculator used for?

The Function Calculator is a tool used to analyze functions. It can find the following for a function: parity, domain, range, intercepts, critical points, intervals of increase/decrease, local and global extrema, concavity intervals, inflection points, derivative, integral, asymptotes, and limit. The calculator will also plot the function's graph.

#### How accurate are the results from this calculator?

You can trust our calculator because it provides accurate results.