Function Calculator

Use this function equation calculator to solve and perform operations on mathematical equations involving functions. Our tool supports a wide range of functions including linear, quadratic, polynomial, exponential, logarithmic, or trigonometric functions. To find a quick and step-by-step solution, you can also upload an image instead of writing or pasting a question manually.

What is a Function Equation?

A function equation expresses a relationship between two or more variables (dependent and independent), where each input is assigned exactly one output according to a specific rule or function.

Standard Form:

A function is generally denoted as f(x) or g(x) instead of using y.

f (x) = y

Where:

x = Input
f = Function
y = Desired function

How To Use Function Notation Calculator?

With the help of our online function calculator, you can get instant solutions and repeat the process for an unlimited number of questions.

Enter the function f(x), copy/paste it, or upload a photo
Click on "Calculate"

How to Calculate Function Equations?

To calculate a function equation, you typically need to follow the below steps:

✔️ Identify the variables:

Find the input variable (usually "x") and output variable (usually "y").

✔️ Find the relationship:

Analyze the given information to find how output changes with input.

✔️ Write the function equation:

Write the function equation by using the slope-intercept form "y = mx + b".

✔️ Other Functions:

Depending on the type of function, use another form such as exponential, logarithmic, or trigonometric.

✔️ Substitute known values:

If you have specific points on a graph or other information, then substitute these into a formula to know the unknown coefficients.

Example:

Find the equation representing a linear function that intersects the points (2, 5) and (4, 11).

Solution:

Step 1: Identify variables

"x" is the input
"y" is the output

Step 2: Find the slope

m = (y2 - y1) ÷ (x2 - x1)

m = (11 - 5) ÷ (4 - 2)

m = 3

Step 3: Use the slope and one point to find the y-intercept:

5 = 3(2) + b

b = -1

Step 4: Write the function equation:

y = 3x - 1

What are the Types of Functions?

There are various fundamental types of functions, each having different properties.

1. Linear Functions

A linear function is a function whose graph is a straight line. It has the general form:

f (x) = mx + b

m = slope of the line
b = y-intercept (the point where the line crosses the y-axis)

2. Quadratic Functions

A quadratic function is a polynomial function of degree 2, with the general form:

f (x) = ax² + bx + c

a, b, and c = constants
a ≠ 0

3. Polynomial Functions

A polynomial function is a function that involves only non-negative integer powers of x. It has the general form:

f (x) = a_n xⁿ + a_n-1 x^n-1 + ... + a₁ x + a₀

n = non-negative integer (the degree of the polynomial).
a_n, a_n-1, ..., a₀ = constants with a_n ≠ 0

4. Rational Functions

A rational function is a ratio of two polynomial functions:

f (x) = p(x) / q(x)

p(x) and q(x) = polynomials
q(x) ≠ 0

5. Exponential Functions

An exponential function involves the variable x in the exponent. It has the general form:

f(x) = a . b^x

a = initial value (the output when x = 0)
b = base, a positive real number.

6. Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It has the general form:

f(x) = log_b (x)

b = base of the logarithm, b > 0 and b ≠ 1
The function answers the question: "To what power must b be raised to obtain x?

7. Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The basic trigonometric functions are:

Sine: sin(x)
Cosine: cos(x)
Tangent: tan (x)

8. Inverse Functions

An inverse function reverses the input-output relationship of a given function, meaning if f(x) = y, then the inverse function f⁻¹(y) = x

x = f⁻¹(y)

9. Greatest Integer Functions

The greatest Integer Function returns the largest integer less than or equal to a given number.

f(x) = ⌊x⌋

Where ⌊x⌋ represents the greatest integer less than or equal to x.

10. Cubic Functions

A cubic function is a polynomial function with a degree of three, meaning the highest power of the variable x is 3.

f(x) = ax³

What are the Features of This Function Value Calculator?

➱ Handles various function types:

Linear, algebraic, constant, cubic, quadratic, polynomial, exponential, logarithmic, trigonometric, and more.

➱ Performs key operations:

Function composition, finding inverses, calculating derivatives, determining domain and range.

➱ Provides step-by-step solutions:

To enhance the understanding, our tool provides step-by-step solutions for each calculation.

➱ Offers image upload functionality:

Easily input equations by uploading images of the problem.

➱ Features a user-friendly interface:

Intuitive design for easy navigation and interaction.

FAQs:

How do you identify a function from an equation?

Step # 1: Find the solution to the equation if needed

Step # 2: Find out how many outputs for any given input

There is a simple way to identify the function is that if there is one or zero output for any input then it is a function otherwise we can say that the equation does not define a function.

What is the general form of a function equation?

The function equation is denoted by f(x) and its general representation is y = f(x).

Can every mathematical relation be represented as a function?

No, all mathematical equations can't be rewritten as a functional formula.