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 Calculator
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 \(a \neq 0\).
Quadratic function
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 \(-3\)
\(g (−3) = (−3)2 + 2 = 9 + 2 = 11\)
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.