Function Notation in Math

Q: What is the domain and range of a function?

Domain: The set of all possible input values (or x-values) for which the function is defined.Range: The set of all possible output values (or y-values) produced by the function.

Last Updated : 10 Feb, 2025

Function notation is a precise and simplified way to express the relationship between inputs and outputs. Instead of using the typical y = format, function notation replaces y with a function name, such as f(x), where f represents the function's name, and x is the input variable. This format helps manage multiple functions and better understand how changes in input affect the output.

For example, in the image above the function f(x) = 35x² + 2, f(x) tells us the output when x is substituted into the equation.

A function is a rule that links two sets of values: the domain (input values) and the range (output values). For each value in the domain, there is exactly one corresponding value in the range.

Mathematically, a function f from set A to set B can be written as:

f : A → B

This means that for every element in the domain A, there is a unique corresponding element in the range B.

Writing Functions Using Notations

Writing functions in notation is a brilliant way of describing functions in mathematics since it is brief. Instead of writing y = 2x + 3, we write f(x) = 2x + 3 Here, f represents an operation that is applied on the quantity x. This is more flexible and better for function manipulation.

In function notation:

f(x) is a notation that is always stated as “f of x” and it stands for the value of the function at x.
The letter f also stands for the name of the function while x is the argument of this function.
The expression inside the parentheses defines the arguments of the function that are taken as the input.

For example, if f(x) = 3x + 5, then:

f(2) = 3(2) + 5 = 6 + 5 = 11
This shows that when x = 2, the output of the function f(x) is 11.

Examples of Function Notation

Function notation can be applied in various contexts:

Linear Function: For f(x) = 4x − 7,

f(3) = 4(3) − 7 = 12 − 7 = 5.

Quadratic Function: For g(x) = x² − 4x + 6,

g(2) = 2² − 4(2) + 6 = 4 − 8 + 6 = 2.

Piecewise Function: $h(x) =\begin{cases} x + 2 & \text{if } x \geq 0 \\-x + 2 & \text{if } x < 0\end{cases}$

h(3) = 3 + 2 = 5
h(-2) = -(-2) + 2 = 4

Common Types of Functions Expressed in Function Notation

Some common types of functions expressed in function notation:

Linear Functions:
- Form: f(x) = mx +b
- Example: f(x) = 2x + 3

Quadratic Functions:
- Form: f(x) = ax²+ bx + c
- Example: f(x) = x²− 4x + 4

Cubic Functions:
- Form: f(x) = ax³+ bx²+ cx + d
- Example: f(x) = 2x³− 3x²+ x − 5

Exponential Functions:
- Form: f(x) = a⋅b^x
- Example: f(x) = 3 ⋅ 2^x

Logarithmic Functions:
- Form: f(x) = a ⋅ log⁡_b(x) + c
- Example: f(x) = 2 ⋅ log⁡2(x) − 1

Trigonometric Functions:
- f(x) = sin⁡(x)
- f(x) = cos⁡(x)
- f(x) = tan⁡(x)

Rational Functions:
- Form: f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.
- Example: f(x) = (x²− 1)/(x + 1)

Piecewise Functions:
- Form: $f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ x + 1 & \text{if } x \geq 0 \end{cases}$

Absolute Value Functions:
- Form: f(x) = ∣x∣
- Example: f(x) = ∣x − 3∣

Solved Questions on Functions Notations

Question 1: Consider the linear function f(x) = 3x − 4. Find f(2) and determine the x-intercept.

Solution:

To find f(2): f(2) = 3(2) − 4 = 6 − 4 = 2
So, f(2) = 2.
To find the x-intercept, set f(x) = 0 and solve for x:
0 = 3x − 4
3x = 4⟹x = 4/3
Thus, the x-intercept is x = 4/3.

Question 2: Given the quadratic function g(x) = 2x² − 3x + 1, find the value of g( − 1) and the vertex of the parabola.

Solution:

To find g( − 1): g( − 1) = 2( − 1)² − 3( − 1) + 1 = 2(1) + 3 + 1 = 6
So, g( − 1) = 6.
The vertex of a quadratic function ax² + bx + c is given by:
Substituting x = 3/4 into g(x):
$g\left(\frac{3}{4}\right) = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 1 = \frac{18}{16} - \frac{36}{16} + \frac{16}{16} = \frac{-2}{16} = -\frac{1}{8}$
The vertex is (3/4, − 1/8).

Question 3: Find the domain of the rational function h(x) = 5/x² − 9.

Solution:

The domain of a rational function is all real numbers except where the denominator is zero:
x² − 9 = 0
(x − 3)(x + 3) = 0
So, x = 3 and x = − 3 make the denominator zero. Thus, the domain is:
x ∈ R ,x = 3,x = − 3
The domain is ( − ∞, − 3)∪( − 3, 3)∪(3, ∞).

Question 4: Evaluate the exponential function f(x) = 2⋅3^x at x = − 2 and determine if the function represents growth or decay.

Solution:

To find f( − 2):
f( − 2) = 2⋅3^{− 2} = 2⋅ 1/9 = 2/9
So, f( − 2) = 2/9.
Since the base b = 3 is greater than 1, the function represents exponential growth.

Practice Questions of Function Notation

Question 1: Linear Function: Given f(x) = 4x + 5, find f( − 3) and determine the y-intercept.

Question 2: Quadratic Function: If g(x) = x² − 4x + 4, find the roots of the equation by factoring.

Question 3: Polynomial Function: For the polynomial p(x) = x³ − 6x² + 11x − 6, find the value of p(1).

Question 4: Exponential Function: Solve for x in the equation 5⋅2^𝑥 = 40.

Question 5: Logarithmic Function: If f(x) = log₂(x − 1), find the domain of f(x).

Question 6: Piecewise Function: For the function h(x) defined as: $h(x) =\begin{cases} 2x + 1 & \text{if } x \geq 0 \\-x^2 + 3 & \text{if } x < 0\end{cases}$

Find h(−3) and h(2).

Question 7: Rational Function: Determine the vertical asymptotes of the function $h(x) = \frac{x^2 + 1}{x^2 - 4}$

Answer Key
f( − 3) = − 7, y-intercept is 5.
Roots are x = 2 (double root).
p(1) = 0.
x = 3.
Domain is x>1.
h( − 3) = − 6, h(2) = 5.
Vertical asymptotes at x = 2 and x = − 2.

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FAQs on Function Notations

What is a function in mathematics?

A function is a relationship between two sets where each element in the first set (domain) is paired with exactly one element in the second set (range). The function assigns every input exactly one output.

How is a function typically notated?

A function is commonly notated as f(x), where f is the function's name and x is the input variable. The expression f(x) represents the value of the function at x.

What is the domain and range of a function?

Domain: The set of all possible input values (or x-values) for which the function is defined.
Range: The set of all possible output values (or y-values) produced by the function.