Comprehensive Guide to Mathematical Functions

Absolute Value or Modulus Function

The absolute value (or modulus) ∣x∣|x| of a real number xx is the non-negative value of xx, irrespective of its sign. It is defined as:∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

Key Points:

• ∣x∣=x|x| = x for x>0x > 0.
• ∣x∣=−x|x| = -x for x<0x < 0.
• ∣0∣=0|0| = 0.

Examples:

• ∣3∣=3|3| = 3
• ∣−3∣=3|-3| = 3
• The absolute value of a number represents its distance from zero.

---

Exponential Function

In mathematics, the exponential function is defined as exe^x, where ee is a mathematical constant approximately equal to 2.718281828. This function has unique properties:

• It is its own derivative: ddxex=ex\frac{d}{dx}e^x = e^x.
• Models situations where a constant change in the independent variable results in a proportional change in the dependent variable.

Common Notations:

• exe^x
• exp(x)\text{exp}(x)

Applications:

• Population growth
• Radioactive decay
• Compound interest

---

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the ratios of its sides. These functions are widely used in geometry, physics, and engineering.

Key Trigonometric Functions:

1. Sine (sin⁡\sin): Opposite side / Hypotenuse
2. Cosine (cos⁡\cos): Adjacent side / Hypotenuse
3. Tangent (tan⁡\tan): Opposite side / Adjacent side

Graphs of sin⁡(x)\sin(x), cos⁡(x)\cos(x), and tan⁡(x)\tan(x):

• sin⁡(x)\sin(x) and cos⁡(x)\cos(x) are periodic with a period of 2π2\pi.
• tan⁡(x)\tan(x) is periodic with a period of π\pi and has vertical asymptotes.

Relation Between Trigonometric Functions:

• sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1
• tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}
• 1+tan⁡2(x)=sec⁡2(x)1 + \tan^2(x) = \sec^2(x)

---