Common Functions

The Function Families

Certain functions appear everywhere in mathematics. Each has a distinctive shape and behavior.

Learning to recognize them helps you understand more complex functions built from these pieces.

Linear Functions

$f(x) = mx + b$

The simplest functions - straight lines.

• m = slope (steepness and direction)
• b = y-intercept (where the line crosses the y-axis)

Key properties:

• Constant rate of change (the slope)
• Graph is always a straight line
• Domain and range are all real numbers

Special case: When $m = 0$, you get a horizontal line $f(x) = b$.

Quadratic Functions

$f(x) = ax^2 + bx + c$

Parabolas - smooth U-shaped curves.

• $a > 0$: opens upward
• $a < 0$: opens downward
• The vertex is the highest or lowest point

Key properties:

• One turning point (the vertex)
• Symmetric about a vertical line through the vertex
• Domain is all real numbers
• Range depends on whether it opens up or down

Polynomial Functions

$f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

Linear and quadratic are special cases. The degree (highest power) determines the shape.

Key properties:

• Smooth, continuous curves
• No breaks, holes, or asymptotes
• End behavior depends on the leading term

Higher degree means more possible “wiggles” in the graph.

Rational Functions

$f(x) = \frac{p(x)}{q(x)}$

One polynomial divided by another.

The simplest example: $f(x) = \frac{1}{x}$

Asymptotes are lines the graph approaches but never touches:

• Vertical asymptote: where the denominator equals zero
• Horizontal asymptote: the value approached as $x \to \pm\infty$

Rational functions can have gaps in their domain where they’re undefined.

Exponential Functions

$f(x) = a^x$

The variable is in the exponent.

• Base $a > 1$: exponential growth
• Base $0 < a < 1$: exponential decay

Key properties:

• Always positive (never touches the x-axis)
• Horizontal asymptote at $y = 0$
• Grows (or decays) faster and faster

The natural exponential $e^x$ (where $e \approx 2.718$) appears throughout calculus and science.

Logarithmic Functions

$f(x) = \log_a(x)$

The inverse of exponential functions.

$\log_a(x)$ asks: “What power of $a$ gives $x$?”

Key properties:

• Only defined for $x > 0$
• Vertical asymptote at $x = 0$
• Grows slowly - very slowly for large $x$

Logarithms turn multiplication into addition, making them useful for very large or very small numbers.

These are the building blocks. Most functions you encounter are combinations or transformations of these families.