Functions

Functions

Functions are mathematical relationships that assign exactly one output to each input. Here's what you need to know:

Basic Concepts

Definition

A function is a rule that assigns to each element in the domain exactly one element in the range.

Function Notation

$$f(x)$$ represents the output of function f when the input is x.

Example: If $$f(x) = 2x + 3$$, then $$f(4) = 2(4) + 3 = 11$$

Domain and Range

• Domain: The set of all possible input values
• Range: The set of all possible output values

Finding the domain: Exclude values that cause:

• Division by zero
• Even roots of negative numbers

Example: For $$f(x) = \frac{1}{x-2}$$, the domain is all real numbers except 2.

Example: For $$g(x) = \sqrt{x+3}$$, the domain is $$x \geq -3$$.

Types of Functions

Linear Functions

$$f(x) = mx + b$$

Graph: Straight line with slope m and y-intercept b

Quadratic Functions

$$f(x) = ax^2 + bx + c$$ (where a ≠ 0)

Graph: Parabola

Polynomial Functions

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

Degree: The highest power of x (n)

Rational Functions

$$f(x) = \frac{P(x)}{Q(x)}$$ where P and Q are polynomials

Exponential Functions

$$f(x) = a \cdot b^x$$ (where b > 0, b ≠ 1)

Logarithmic Functions

$$f(x) = \log_b(x)$$ (where b > 0, b ≠ 1)

Function Operations

Basic Operations

For functions f and g:

• Sum: $$(f + g)(x) = f(x) + g(x)$$
• Difference: $$(f - g)(x) = f(x) - g(x)$$
• Product: $$(f \cdot g)(x) = f(x) \cdot g(x)$$
• Quotient: $$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$$ (where g(x) ≠ 0)

Composition of Functions

$$(f \circ g)(x) = f(g(x))$$

Example: If $$f(x) = x^2$$ and $$g(x) = x + 3$$, then:

$$(f \circ g)(x) = f(g(x)) = f(x + 3) = (x + 3)^2 = x^2 + 6x + 9$$

$$(g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 3$$

Function Properties

Even and Odd Functions

• Even function: $$f(-x) = f(x)$$ for all x in the domain
• Graph is symmetric about the y-axis
• Example: $$f(x) = x^2$$
• Odd function: $$f(-x) = -f(x)$$ for all x in the domain
• Graph is symmetric about the origin
• Example: $$f(x) = x^3$$

Increasing and Decreasing Functions

• Increasing: If $$x_1 < x_2$$ implies $$f(x_1) < f(x_2)$$
• Decreasing: If $$x_1 < x_2$$ implies $$f(x_1) > f(x_2)$$

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input.

Horizontal Line Test: A function is one-to-one if no horizontal line intersects its graph more than once.

Inverse Functions

Definition

If f is a one-to-one function, its inverse function $$f^{-1}$$ satisfies:

$$f^{-1}(f(x)) = x$$ for all x in the domain of f

$$f(f^{-1}(x)) = x$$ for all x in the domain of $$f^{-1}$$

Finding the Inverse

1. Replace f(x) with y
2. Interchange x and y
3. Solve for y
4. Replace y with $$f^{-1}(x)$$

Example: Find the inverse of $$f(x) = 3x - 5$$

Step 1: y = 3x - 5

Step 2: x = 3y - 5

Step 3: x + 5 = 3y, so y = (x + 5)/3

Step 4: $$f^{-1}(x) = \frac{x + 5}{3}$$

Graph of the Inverse

The graph of $$f^{-1}$$ is the reflection of the graph of f across the line y = x.

Common Pitfalls and Tips

• Not every relation is a function—check the vertical line test
• Not every function has an inverse function—check the horizontal line test
• Domain restrictions may apply when finding inverses
• Be careful with notation: $$f^{-1}(x)$$ means the inverse function, not 1/f(x)
• When composing functions, work from the inside out
• Remember that function composition is not generally commutative: $$f \circ g ≠ g \circ f$$