Function notation solver calculator

Function notation solver calculator DEFAULT

Function Composition

Function Composition Calculator

Function Inputs & Outputs

When you see something like f(g(x)) or (f∘g)x, you have function composition.

First off, think of handling functions like doing a find & replace.

When we have a function like f(x)=4x2+6x+5, plugging an input in for x means finding & replacing every x with the input value. For example, if our input is 3, we find & replace to get:




And you have to make sure you replace every x. It's like how they had to replace every creepy Sonic with the new, cute Sonic, otherwise the creepiness would still be there ☠️.

If you need more help with functions, check out our functions lesson and calculator here!

What is Function Composition?

The idea behind function composition is that instead of a number, another function acts as the input.

So, let's say we have these two functions:


If we're asked for f(g(x)), then our input is g(x), and we need to replace every x in f(x) with g(x):




And since we know that g(x)=2x+1, we can simplify f(g(x)) by using 2x+1 instead of g(x) on the right side:


And that's how we get f(g(x))!

Plugging in a Value

Now, if we're asked for f(g(3)) instead of f(g(x)), then we need to do another find & replace to replace every x in our new f(g(x)) equation with our input value of 3:




Then we have all numbers on the right side of the equation, and we can just calculate to find our final answer of f(g(3))=

Graphing Function Composition

You might get the functions in graph forms, like this:

In this case, we use our graphs to find the function composition. Let's say we're asked for f(g(1)).

This means, first we need to find what g(1) is. Let's look at our graph for g(x) where our x=1:

Looking at the point on the line where x=1, we can see that the y value is 2. This means that g(1)=2.

Now, if we go back to our original composition of f(g(1)), we can replace g(1) with 2 to get f(g(1))=f(2).

Now, we just look at our graph of f(x) at the point where x=2:

Looking at the point on the line where x=2, we can see that the y value is 4. This means that f(2)=4.

So to review, to solve for f(g(1)), we first needed to find the value of g(1). Once we found that g(1)=2, we replaced g(1) with 2 to get:


Looking at the graph for f(x), we find:


Function Composition through Graphs

Let's say we're looking for f(g(3)).

Step 1: Find the inner value.

First, we use the graph for g(x) to find the value of g(3).

Step 2: Plug in the inner value.

Next, we plug in the value we found into the composition. Let's say we found that our value was 2. We plug that into our composition to replace g(3) to get f(g(3))=f(2).

Step 3: Find the final value.

Finally, we use the graph for f(x) to find the value of f(2). Once we do that, we're done!


Function table (2 variables) Calculator

[1]  /09/30    20 years old level / An office worker / A public employee / Very /

Purpose of use
To create a function for a python script where it inversely counted down or up on each column by rows.
Super helpful thank you!

[2]  /09/09    Under 20 years old / High-school/ University/ Grad student / A little /

Purpose of use
trying to find function using a table without an equation

[3]  /08/16    Under 20 years old / High-school/ University/ Grad student / Very /

Purpose of use
Function Purposes

[4]  /07/02    Under 20 years old / High-school/ University/ Grad student / Not at All /

Purpose of use
I needed to find the pattern for [harder] linear functions quickly to move on with equation faster/ more accurately
something more clear cut

[5]  /05/03    Under 20 years old / Elementary school/ Junior high-school student / Not at All /

Purpose of use
To find the equation needed with a table
Directions need to be more precise and there should be more options on if we want the equation or what this is.

[6]  /04/27    Under 20 years old / High-school/ University/ Grad student / Not at All /

Purpose of use
To have my calculations be faster for my homework.

[7]  /04/25    Under 20 years old / Elementary school/ Junior high-school student / Not at All /

Purpose of use
To finish table of parabolas math sheet
Less requirements for the calculator, I want to find the y not the expression

[8]  /04/20    Under 20 years old / Elementary school/ Junior high-school student / A little /

Purpose of use
Need help with math homework.
Need to have the equation of the function table.

[9]  /03/18    Under 20 years old / Elementary school/ Junior high-school student / A little /

Purpose of use
Just practice
I need to determine what function (linear, quadratic, or exponential) functions from tables.

[10]  /03/15    Under 20 years old / High-school/ University/ Grad student / Very /

Purpose of use
Determining if a function is linear or nonlinear.

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Most Used Actions

\mathrm{domain} \mathrm{range} \mathrm{inverse} \mathrm{extreme\:points} \mathrm{asymptotes}
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Solver function calculator notation

Function Calculator

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.

Your input: find the properties of $$$f=x^{3} - 3 x^{2}$$$


The function is neither even nor odd.


$$$\left(-\infty, \infty\right)$$$







$$$\left(-\infty, \infty\right)$$$

Critical Points

$$$\left(x, f \left(x \right)\right)=\left(0,0\right)$$$

$$$\left(x, f \left(x \right)\right)=\left(2,-4\right)$$$

Intervals of Increase

$$$\left(-\infty, 0\right) \cup \left(2, \infty\right)$$$

Intervals of Decrease

$$$\left(0, 2\right)$$$

Local Minima

$$$\left(x, f \left(x \right)\right)=\left(2,-4\right)$$$

Local Maxima

$$$\left(x, f \left(x \right)\right)=\left(0,0\right)$$$

Global (Absolute) Minima and Maxima

For global minima and maxima, see extrema calculator.

Inflection Points

$$$\left(x, f \left(x \right)\right)=\left(1,-2\right)$$$

Concave upward on

$$$\left(1, \infty\right)$$$

Concave downward on

$$$\left(-\infty, 1\right)$$$


For derivative, see derivative calculator.


For integral, see integral calculator.


For asymptotes, see asymptote calculator.


For limit, see limit calculator.

Taylor Polynomial

For Taylor polynomial, see taylor polynomial calculator.


For graph, see graphing calculator.

❖ Function Notation ❖

Function Notation . . . f(x)

Background -why the Function Notation symbol is used . . .

Symbols are used as aconcise way of giving lengthy instructions

related to numbers and logic.

Math Symbols are a communication tool. Symbols are an invention which is used to eliminate the need to write long, plain language instructions to describe calculations and other processes.

The most valuable, most frequently used Symbols in
mathematics . . .

The most important, most frequently used Miscellaneous symbols are listed below.

Math - Symbol for Function of xMath Notation for the "Function" of x -

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The Function Notation f(x) means two things:

(1)f(x) literally means that the value of f(x) depends upon the value of x.

And . . .

(2) For every value of x, there is only one unique value f(x).

f(x) is often used interchangeably with y.

Most of the time this is OK, but y is not the same as f(x).

f(x) is a function. f(x) always satisfies both of the criteria listed as (1) and (2), shown above.

Y may or may not represent a function. It may only satisfy the criteria listed as (1).

Example 1:   f(x), function of x (can be used interchangeably with the y in this example)

y = x + 1

f(x) = x + 1

if x = 4

y = 4 + 1

y = 5

f(4) = 4 + 1

f(4) = 5

Note: both y and f(x) represent a function. Both notations satisfy the criteria for (1) and (2) above.

Note: both y
and f(x) = 5.

Example 2:   y, the following relation is not a function

Not a Function

Not a Function

Note: there are two values of y for every value of x. A function can have only one value of y for each value of x. Therefore, this is not a function.


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Online Domain and Range Calculator


Find the domain and range of a function with Wolfram|Alpha

More than just an online function properties finder

Wolfram|Alpha is a great tool for finding the domain and range of a function. It also shows plots of the function and illustrates the domain and range on a number line to enhance your mathematical intuition.

Learn more about:

Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for the domain and range.

What is domain and range?

The domain of a function, , is most commonly defined as the set of values for which a function is defined. For example, a function that is defined for real values in has domain , and is sometimes said to be "a function over the reals." The set of values to which is sent by the function is called the range.

Informally, if a function is defined on some set, then we call that set the domain. The values taken by the function are collectively referred to as the range. For example, the function takes the reals (domain) to the non-negative reals (range). The sine function takes the reals (domain) to the closed interval (range). (Both of these functions can be extended so that their domains are the complex numbers, and the ranges change as well.)


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