Functions

A function is a rule that assigns each input exactly one output.

• Think of it like a machine: you feed it an input (say, $x$), and it processes it to give you one output $f(x)$.
• Every input must have only one output. If an input gives two different outputs, it’s not a function.

Notation:

• $f(x)$ is read as “f of x”
• $f(x) = x^2 + 3x + 1$ means: input $x$ gives output $x^2 + 3x + 1$

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Key Terminology

Example:
For $f(x) = \sqrt{x - 2}$,

• Domain: $x \geq 2$ (because square root must be real)
• Range: $f(x) \geq 0$

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Types of Functions

1. Identity Function

$$f(x) = x \quad \text{(output equals input)}$$

2. Constant Function

$$f(x) = c \quad \text{(same output for any input)}$$

3. Linear Function

$$f(x) = ax + b \quad \text{(graph is a straight line)}$$

4. Quadratic Function

$$f(x) = ax^2 + bx + c \quad \text{(graph is a parabola)}$$

5. Cubic, Polynomial Functions

$$f(x) = ax^n + \dots \quad \text{(n = 3 or higher)}$$

6. Rational Function

f(x) = \frac{P(x)}{Q(x)} \quad \text{(P and Q are polynomials, Q(x) \ne 0)}

7. Modulus Function

$$f(x) = |x| \quad \text{(distance from zero)}$$

8. Signum Function

$$f(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases}$$

9. Greatest Integer Function (Floor Function)

$$f(x) = \lfloor x \rfloor \quad \text{(largest integer \le x)}$$

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Important Operations on Functions

1. Function Composition

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

Think of it as nested processing: first apply $g$, then apply $f$ to that result.

2. Inverse Function

• If $f(x) = y$, then $f^{-1}(y) = x$
• Not all functions have inverses. A function must be one-to-one (injective) to have an inverse.

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Function Properties (Must-Know)

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Visual Explanations

• Vertical Line Test: To check if a graph is a function → If any vertical line cuts the graph more than once, it’s not a function.

• Even vs. Odd functions:

  • Even: Parabola $f(x) = x^2$
  • Odd: Cube function $f(x) = x^3$

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Conceptual Tips and Mistakes to Avoid

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Examples

Example 1: Find domain of $f(x) = \frac{1}{x-2}$

• Denominator can’t be zero → $x \neq 2$
• Domain: $x \in \mathbb{R}, x \neq 2$

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Example 2: If $f(x) = x^2$, find $f(-3)$

$$f(-3) = (-3)^2 = 9$$

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Example 3: Is $f(x) = \sqrt{x+1}$ defined for $x = -2$?

$$f(-2) = \sqrt{-2 + 1} = \sqrt{-1} \rightarrow \text{Not real} \Rightarrow \text{Not defined}$$

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Example 4: Composition

Let $f(x) = 2x+1$, $g(x) = x^2$. Find $(f \circ g)(x)$

$$(f \circ g)(x) = f(g(x)) = f(x^2) = 2x^2 + 1$$

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Example 5: Greatest Integer Function

Find $\lfloor -1.3 \rfloor$
Answer: $-2$