# Function Calculator

The Function Calculator is a free online tool that displays the graph of a given function. With STUDYQUERIES’s online function calculator tool, the calculation is faster and displays the graph of the function by calculating the x and y-intercept values, slope values in a fraction of a second.

## How to Use a Function Calculator?

To use the function calculator, follow these steps:

• Step 1: Enter the function f(x) in the appropriate input field
• Step 2: Click on “Graph” to get the results
• Step 3: The graph of the function will appear in a new window

## Function Notation And Calculator– Explanation & Examples

Functions were developed in the seventeenth century by Rene Descartes, who used them to model mathematical relationships in his book Geometry. Fifty years after the publication of Geometry, Gottfried Wilhelm Leibniz introduced the term “function.”

Later, Leonhard Euler formalized the usage of functions when he introduced the concept of function notation; $$\color{red}{y = f (x)}$$. It was until 1837 when Peter Dirichlet – a German mathematician gave the modern definition of a function.

### Definition Of Function

The definition of a function is that it is a set of inputs with a single output in each case. All functions have a domain and range. The domain is the set of independent values of the variable $$x$$ for a relationship or a function is defined. In simple words, the domain is a set of $$x-values$$ that generate the $$real\ values\ of\ y$$ when substituted in the function.

In contrast, the range is a set of all possible values a function can produce. An interval notation or inequalities can be used to express the range of a function.

### What is a Function Notation?

In notation, elements such as phrases, numbers, words, etc. are represented by symbols or signs.

Therefore, function notation is a way to represent a function using symbols and signs. Using function notation, a function can be described more simply without a lengthy explanation.

The most frequently used function notation is $$f(x)$$ which is read as $$f\ of \ x$$. In this case, the letter $$x$$, placed within the parentheses and the entire symbol $$f(x)$$, stand for the domain set and range set respectively.

While f is the most popular letter when writing function notation, any other letter of the alphabet can also be used either in upper or lower case.

### Advantages Of Using Function Notation

• Since most functions are represented with various variables such as; $$a, f, g, h, k, etc.$$, we use $$f(x)$$ in order to avoid confusion as to which function is being evaluated.
• Function notation allows identifying the independent variable with ease.
• Function notation also helps us to identify the elements of a function that has to be examined.

Consider a linear function $$y = 3x + 7$$. To write such function in function notation, we simply replace the variable $$y$$ with the phrase $$f(x)$$ to get;

$$f(x) = 3x + 7$$

This function $$f(x) = 3x + 7$$ is read as the $$value\ of\ f\ at\ x$$ or as $$f\ of\ x$$.

### Types Of Functions

There are several types of functions in Algebra. The most common types of functions include:

#### Linear function

A linear function is a polynomial of the first degree. A linear function has the general form of

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

where $$a$$ and $$b$$ are numerical values and $$a \neq 0$$.

A polynomial function of the second degree is known as a quadratic function. The general form of a quadratic function is

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

where $$a, b\ and\ c$$ are integers and $$a \neq 0$$.

#### Cubic function

This is a polynomial function of $$3rd\ degree$$ which is of the form

$$f(x) = ax^3 + bx^2 + cx + d$$

#### Logarithmic function

A logarithmic function is an equation in which a variable appears as an argument of a logarithm. The general of the function is

$$f(x)=log_{a}(x)$$

where $$a$$ is the base and $$x$$ is the argument.

#### Exponential function

An exponential function is an equation in which the variable appears as an exponent. Exponential function is represented as

$$f(x) = a^x$$

#### Trigonometric function

$$f(x) = sinx,\ f(x) = cosx\ etc$$

are examples of trigonometric functions.

Identity Function:

An identity function is such that

$$f: A\rightarrow B\ and\ f(x) = x, \forall\ x \in A$$

Rational Function:

A function is said to be rational if $$R(x) = \frac{P(x)}{Q(x)},\ where\ Q(x) \neq 0.$$

### How to Evaluate Functions?

Function evaluation is the process of determining the output values of a function. This is done by substituting the input values in the given function notation.

$$\pmb{\color{red}{Write\ y = x^2 + 4x + 1\ using\ function\ notation\ and\ evaluate\ the\ function\ at\ x = 3.}}$$

Given, $$y = x^2 + 4x + 1$$

By applying function notation, we get

$$f(x) = x^2 + 4x + 1$$

Substitute $$x$$ with $$3$$

$$f (3) = 32 + 4 \times 3 + 1 = 9 + 12 + 1 = 22$$

$$\pmb{\color{red}{Evaluate\ the\ function\ f(x) = 3(2x+1)\ when\ x = 4.}}$$

Plug $$x = 4$$ in the function $$f(x)$$.

$$f (4) = 3(2(4) + 1)$$

$$f (4) = 3(8 + 1)$$

$$f (4) = 3 \times 9$$

$$f (4) = 27$$

$$\pmb{\color{red}{Write\ the\ function\ y = 2x^2 + 4x – 3\ in\ function\ notation\ and\ find\ f (2a + 3).}}$$

$$y = 2x^2 + 4x – 3 \Rightarrow f (x) = 2×2 + 4x – 3$$

Substitute $$x$$ with $$(2a + 3)$$.

$$f (2a + 3) = 2(2a + 3)2 + 4(2a + 3) – 3$$

$$= 2(4a2 + 12a + 9) + 8a + 12 – 3$$

$$= 8a2 + 24a + 18 + 8a + 12 – 3$$

$$= 8a2 + 32a + 27$$

$$\pmb{\color{red}{Represent\ y = x^3 – 4x\ using\ function\ notation\ and\ solve\ for\ y\ at\ x = 2.}}$$

Given the function $$y = x^3 – 4x$$, replace $$y$$ with $$f(x)$$ to get;

$$f(x) = x^3 – 4x$$

Now evaluate $$f(x)$$ when $$x = 2$$

$$\Rightarrow f (2) = 23 – 4 \times 2 = 8 -8 = 0$$

Therefore, the value of $$y$$ at $$x=2$$ is $$0$$

$$\pmb{\color{red}{Find\ f (k + 2)\ given\ that,\ f(x) = x^2 + 3x + 5.}}$$

To evaluate f (k + 2), substitute x with (k + 2) in the function.

$$\Rightarrow f (k + 2) = (k + 2)^2 + 3(k + 2) + 5$$

$$\Rightarrow k^2 + 2^2 + 2k (2) + 3k + 6 + 5$$

$$\Rightarrow k^2 + 4 + 4k + 3k + 6 + 5$$

$$= k^2 + 7k + 15$$

$$\pmb{\color{red}{Given\ the\ function\ notation\ f (x) = x^2 – x – 4.\ Find\ the\ value\ of\ x\ when\ f (x) = 8}}$$

$$f (x) = x^2 – x – 4$$

Substitute $$f(x)$$ by $$8$$.

$$8 = x^2 – x – 4$$

$$x^2 – x – 12 = 0$$

Solve the quadratic equation by factoring to get;

$$\Rightarrow (x – 4) (x + 3) = 0$$

$$\Rightarrow x – 4 = 0;\ x + 3 = 0$$

Therefore, the values of $$x$$ when $$f (x) = 8$$ are;

$$x = 4;\ x = -3$$

$$\pmb{\color{red}{Evaluate\ the\ function\ g(x) = x^2 + 2\ at\ x = −3}}$$

Substitute $$x$$ with $$-3$$

$$g (−3) = (−3)2 + 2 = 9 + 2 = 11$$

## Composite Function Calculator:

A composite function calculator is a tool that helps in evaluating composite functions. A composite function is formed by combining two or more functions, where the output of one function becomes the input for another. The calculator allows you to input the functions and the values of the variables to compute the composite function’s result.

Example: Given f(x) = 2x and g(x) = x^2, the composite function (f ∘ g)(x) is calculated by substituting the expression g(x) into f(x). So, (f ∘ g)(x) = f(g(x)) = 2(g(x)) = 2(x^2) = 2x^2.

## Piecewise Function Calculator:

A piecewise function calculator helps in evaluating and graphing piecewise functions. A piecewise function is defined by different rules or formulas for different intervals or ranges of the input variable. The calculator allows you to input the different rules and determine the output values for specific input values or intervals.

Example: Consider the piecewise function:
f(x) = {
x + 1, if x < 0,
2x, if x ≥ 0
}
If we want to evaluate f(2), we use the second rule of the function since 2 is greater than or equal to 0. Therefore, f(2) = 2(2) = 4.

## Exponential Function Calculator:

An exponential function calculator helps in evaluating exponential functions. Exponential functions are of the form f(x) = a^x, where a is a constant and x is the variable. The calculator allows you to input the base (a) and the exponent (x) to calculate the value of the exponential function.

Example: For f(x) = 2^x, if we want to evaluate f(3), we substitute x with 3 in the function: f(3) = 2^3 = 8.

## Linear Function Calculator:

A linear function calculator helps in evaluating linear functions. Linear functions are of the form f(x) = mx + b, where m is the slope and b is the y-intercept. The calculator allows you to input the values of the slope and y-intercept, as well as the variable value, to determine the output of the linear function.

Example: For f(x) = 3x + 2, if we want to find f(4), we substitute x with 4 in the function: f(4) = 3(4) + 2 = 14.

## Domain of a Function Calculator:

A domain of a function calculator helps in determining the domain of a given function. The domain of a function represents the set of all possible input values (x-values) for which the function is defined. The calculator allows you to input the function and calculates the domain based on any restrictions or limitations on the variable.

Example: For f(x) = √(x – 5), the domain would be all x-values greater than or equal to 5, since the square root is not defined for negative values.

## Function Calculator with Steps:

A function calculator with steps provides a detailed explanation of the steps involved in solving or evaluating a given function. It breaks down the calculations and shows the intermediate steps to help understand the process.

Example: When using a function calculator with steps to evaluate f(x) = 2x + 3 for x = 4, it would provide the detailed calculation, such as substituting x with 4 in the function: f(4) = 2(4) + 3 = 8 + 3 = 11.

## Find f(x) Calculator:

A “find f(x)

calculator” is used to determine the value of a function at a specific input or variable value. It helps in evaluating the output of the function for a given input value.

Example: Suppose we have the function f(x) = 3x^2 – 2x + 5. If we want to find f(2), we substitute x with 2 in the function: f(2) = 3(2)^2 – 2(2) + 5 = 12.

## Function Notation Calculator:

A function notation calculator helps in performing calculations or operations involving functions using the proper function notation. It allows you to input the function, variable values, and operations to obtain the desired results.

Example: If we have the function f(x) = 4x – 3, and we want to calculate f(x + 2), we substitute x + 2 in place of x in the function: f(x + 2) = 4(x + 2) – 3 = 4x + 8 – 3 = 4x + 5.

## Function Calculator Table:

A function calculator table generates a table of values for a given function. It allows you to input the function and define the range or interval of x-values. The calculator then calculates the corresponding y-values and presents them in a table format.

Example: For the function f(x) = x^2, a function calculator table may generate values like:

x | f(x)
———
-2 | 4
-1 | 1
0 | 0
1 | 1
2 | 4

## Evaluate the Function Calculator:

An evaluate the function calculator allows you to compute the output of a given function for a specific input value. It takes the function and the input value and calculates the corresponding output.

Example: If we have the function f(x) = 2x – 5 and want to evaluate f(3), we substitute x with 3: f(3) = 2(3) – 5 = 6 – 5 = 1.

## FAQs

What is an example of function notation?

Consider a linear function y = 3x + 7. To write such function in function notation, we simply replace the variable y with the phrase f(x) to get; f(x) = 3x + 7. This function f(x) = 3x + 7 is read as the value of f at x or as f of x.

How do you write a function notation?

An equation involving x and y, which is also a function, can be written in the form y = “some expression involving x”; that is, y = f ( x). This last expression is read as “ y equals f of x” and means that y is a function of x.

What is the function equation?

Functional equations are equations where the unknowns are functions, rather than a traditional variable. Each functional equation provides some information about a function or about multiple functions. For example, f ( x ) − f ( y ) = x − y f(x)-f(y)=x-y f(x)−f(y)=x−y is a functional equation.

What does a function equation look like?

For example, in the equation “3 = x – 4,” x = 7. However, a function is an equation in which all of the variables are dependent upon the independent numbers in the mathematical statement. For instance, in the function “2x = y,” y is dependent upon the value of x to determine its numerical worth.

How do you find a function?

Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function.

How To Find The Domain Of A Function Calculator?

To find the domain of a function using a calculator, you need to input the function into a suitable calculator that supports domain calculations. The calculator will then analyze the function and provide the domain as the set of all valid input values for the function. The domain can be determined by considering any restrictions or limitations on the variable within the function.

How To Find Zeros Of A Function Calculator?

To find the zeros of a function (also known as the x-intercepts or roots), you can use a calculator specifically designed for finding zeros. Enter the function into the calculator, and it will calculate the values of x where the function equals zero. The calculator may utilize numerical methods or algorithms to approximate the zeros of the function accurately.

How To Find The Inverse Of A Function Calculator?

Finding the inverse of a function can be done using a calculator capable of calculating inverses. Input the original function into the calculator, and it will compute the inverse function for you. Note that not all functions have inverses, so it’s important to ensure that the original function is invertible before using the calculator.

How To Find The Zeros Of A Function Calculator?

The process of finding the zeros of a function is similar to finding the x-intercepts or roots. You can use a calculator with a specific feature for finding zeros. Input the function into the calculator, and it will determine the values of x where the function equals zero, indicating the zeros or roots of the function.

How To Find Inverse Function Calculator?

To find the inverse of a function using a calculator, you need to input the original function into a calculator capable of computing inverses. The calculator will then calculate the inverse function for you. Remember that not all functions have inverses, so it’s important to ensure the original function is invertible before using the calculator. The inverse function calculator will typically output the inverse function, which can be useful for further analysis or computations.