Animated Graph of Function

📘 Mathematics-I

Interactive notes on mathematical functions, their domains, ranges, properties and animated graphs.

Functions and Their Graphs

A function describes a relationship between an input x and an output f(x). Different functions produce characteristic curves.

Click any function below to expand the section. The graph will then be drawn automatically.

1. Exponential Functions

f(x) = e^x

Domain ℝ

Co-domain ℝ

Range (0, ∞)

The value of e^x is always positive. The x-axis is a horizontal asymptote.

f(x) = 2^x

Domain ℝ

Co-domain ℝ

Range (0, ∞)

The graph passes through (0,1) and increases rapidly as x increases.

2. Trigonometric Functions

f(x) = sin x

Domain ℝ

Co-domain ℝ

Range [-1,1]

The sine function has period 2π and oscillates between −1 and 1.

f(x) = cos x

Domain ℝ

Co-domain ℝ

Range [-1,1]

The cosine function has period 2π. Its maximum value is 1 and minimum value is −1.

f(x) = tan x

Domain ℝ \ {π/2 + nπ}

Co-domain ℝ

Range ℝ

Undefined at:
x = π/2 + nπ, where n ∈ ℤ.

3. Hyperbolic Functions

f(x) = sinh x = (e^x − e^−x)/2

Domain ℝ

Co-domain ℝ

Range ℝ

Properties:
sinh(−x) = −sinh x
It is an odd function and passes through (0,0).

f(x) = cosh x = (e^x + e^−x)/2

Domain ℝ

Co-domain ℝ

Range [1,∞)

Properties:
cosh(−x) = cosh x
It is an even function.
Its minimum value is 1 at x = 0.

f(x) = tanh x = (e^x − e^−x) / (e^x + e^−x)

Domain ℝ

Co-domain ℝ

Range (−1,1)

Properties:
tanh(−x) = −tanh x
It is an odd function.
Horizontal asymptotes are y = 1 and y = −1.

4. Logarithmic Function

f(x) = log x

Domain (0,∞)

Co-domain ℝ

Range ℝ

The logarithmic function is defined only for x > 0. The y-axis is a vertical asymptote.

5. Polynomial Function

f(x) = x² − 4x + 3

Domain ℝ

Co-domain ℝ

Range [−1,∞)

x² − 4x + 3 = (x − 2)² − 1

Therefore the vertex is (2,−1).

6. Modulus Function

f(x) = |x|

Domain ℝ

Co-domain ℝ

Range [0,∞)

|x| = x, if x ≥ 0
|x| = −x, if x < 0

Mathematics-I
Interactive Functions • Domain • Range • Properties • Animated Graphs