Library of Functions

Linear Function: $$ f(x) = x $$

Domain: $ (-\infty, \infty) $, all real numbers

Range: $ (-\infty, \infty) $, all real numbers

X-intercept: $ x = 0 $

Y-intercept: $ y = 0 $

Symmetry: Odd function (symmetrical about origin)

Increasing/Decreasing: Always increasing

Local Maxima/Minima: None

Absolute Maxima/Minima: None

Quadratic Function: $$ f(x) = x^2 $$

Domain: $ (-\infty, \infty) $

Range: $ [0, \infty) $

X-intercept: $ x = 0 $

Y-intercept: $ y = 0 $

Symmetry: Even (symmetric about y-axis)

Increasing/Decreasing: Decreasing on $ (-\infty, 0) $, increasing on $ (0, \infty) $

Local Maxima/Minima: Local and absolute minimum at $ f(0) = 0 $

Absolute Maxima/Minima: Absolute minimum at $ f(0) = 0 $, no absolute maximum

Cubic Function: $$ f(x) = x^3 $$

Domain: $ (-\infty, \infty) $

Range: $ (-\infty, \infty) $

X-intercept: $ x = 0 $

Y-intercept: $ y = 0 $

Symmetry: Odd

Increasing/Decreasing: Always increasing

Local Maxima/Minima: None

Absolute Maxima/Minima: None

Absolute Value Function: $$ f(x) = |x| $$

Domain: $ (-\infty, \infty) $, all real numbers

Range: $ [0, \infty) $

X-intercept: $ x = 0 $

Y-intercept: $ y = 0 $

Symmetry: Even (symmetric about y-axis)

Increasing/Decreasing: Decreasing on $ (-\infty, 0) $, increasing on $ (0, \infty) $

Local Maxima/Minima: Local and absolute minimum at $ f(0) = 0 $

Absolute Maxima/Minima: Absolute minimum at $ f(0) = 0 $, no absolute maximum

Square Root Function: $$ f(x) = \sqrt{x} $$

Domain: $ [0, \infty) $, non-negative real numbers

Range: $ [0, \infty) $

X-intercept: $ x = 0 $

Y-intercept: $ y = 0 $

Symmetry: Neither even nor odd

Increasing/Decreasing: Always increasing on $ (0, \infty) $

Local Maxima/Minima: None

Absolute Maxima/Minima: Absolute minimum at $ f(0) = 0 $, no absolute maximum

Cube Root Function: $$ f(x) = \sqrt[3]{x} $$

Domain: $ (-\infty, \infty) $

Range: $ (-\infty, \infty) $

X-intercept: $ x = 0 $

Y-intercept: $ y = 0 $

Symmetry: Odd

Increasing/Decreasing: Always increasing

Local Maxima/Minima: None

Absolute Maxima/Minima: None

Reciprocal Function: $$ f(x) = \frac{1}{x} $$

Domain: $ (-\infty, 0) \cup (0, \infty) $, all real numbers except 0

Range: $ (-\infty, 0) \cup (0, \infty) $

X-intercept: None

Y-intercept: None

Symmetry: Odd (symmetrical about origin)

Increasing/Decreasing: Decreasing on $ (-\infty, 0) \cup (0, \infty) $

Local Maxima/Minima: None

Absolute Maxima/Minima: None

Exponential Function: $$ f(x) = e^x $$

Domain: $ (-\infty, \infty) $

Range: $ (0, \infty) $

X-intercept: None

Y-intercept: $ y = 1 $

Symmetry: Neither even nor odd

Increasing/Decreasing: Always increasing

Local Maxima/Minima: None

Absolute Maxima/Minima: None

Logarithmic Function: $$ f(x) = \ln(x) $$

Domain: $ (0, \infty) $

Range: $ (-\infty, \infty) $

X-intercept: $ x = 1 $

Y-intercept: None

Symmetry: Neither even nor odd

Increasing/Decreasing: Always increasing on $ (0, \infty) $

Local Maxima/Minima: None

Absolute Maxima/Minima: None