1.2 Functions

Detailed Theory: Functions

1. Basic Concepts and Definitions of Functions

1.1 What is a Function?

A function is a special type of relation between two sets where each element of the first set (domain) is associated with exactly one element of the second set (codomain).

Formal Definition: A function $f$ from set A to set B is a rule that assigns to each element $x \in A$ a unique element $y \in B$.

Notation: $f: A \to B$ Read as: "f is a function from A to B"

Visual Representation:

1.2 Key Components of a Function

a) Domain

The set of all possible input values for which the function is defined.

Notation: $\text{Dom}(f)$ or $D_f$

Examples:

1. For $f(x) = \sqrt{x}$, domain = $\{x: x \ge 0\}$ or $[0, \infty)$

2. For $g(x) = \frac{1}{x}$, domain = $\{x: x \neq 0\}$ or $\mathbb{R} - \{0\}$

3. For $h(x) = \log(x)$, domain = $\{x: x > 0\}$ or $(0, \infty)$

b) Codomain

The set of all possible output values. This is the set B in $f: A \to B$.

Important Distinction: Codomain ≠ Range

• Codomain: All values that could possibly come out

• Range: Only the values that actually come out

c) Range/Image

The set of actual output values produced by the function.

Notation: $\text{Range}(f)$ or $\text{Im}(f)$ or $R_f$

Definition: $\text{Range}(f) = \{f(x): x \in \text{Domain}\} \subseteq \text{Codomain}$

Examples:

1. For $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$

   • Domain: $\mathbb{R}$

   • Codomain: $\mathbb{R}$

   • Range: $[0, \infty)$ (only non-negative numbers)

2. For $g: \mathbb{R} \to \mathbb{R}$, $g(x) = \sin x$

   • Domain: $\mathbb{R}$

   • Codomain: $\mathbb{R}$

   • Range: $[-1, 1]$

1.3 Function Notation

• $f(x)$: Value of function f at x (read as "f of x")

• $y = f(x)$: y is the output when input is x

• $f: x \mapsto y$: f maps x to y

Example: For $f(x) = 2x + 3$

• $f(1) = 2(1) + 3 = 5$

• $f(a) = 2a + 3$

• $f(x+h) = 2(x+h) + 3 = 2x + 2h + 3$

1.4 Vertical Line Test

A relation is a function if and only if every vertical line intersects its graph at most once.

Why it works: If a vertical line intersects the graph at more than one point, it means for a single x-value, there are multiple y-values, violating the definition of a function.

Examples:

1. $y = x^2$ passes vertical line test → Function

2. $x^2 + y^2 = 1$ (circle) fails vertical line test → Not a function

3. $y = \sqrt{x}$ passes vertical line test → Function

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2. Types of Functions

2.1 One-to-One (Injective) Function

A function where different inputs give different outputs.

Formal Definition: $f: A \to B$ is injective if $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ for all $x_1, x_2 \in A$

Alternative Definition: $f$ is injective if $x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$ for all $x_1, x_2 \in A$

Horizontal Line Test:

A function is one-to-one if and only if every horizontal line intersects its graph at most once.

Examples:

1. $f(x) = 2x + 3$ is one-to-one Proof: If $2x_1 + 3 = 2x_2 + 3$, then $2x_1 = 2x_2$, so $x_1 = x_2$

2. $g(x) = x^2$ is NOT one-to-one on $\mathbb{R}$ Counterexample: $g(2) = 4$ and $g(-2) = 4$, but $2 \neq -2$

3. $h(x) = e^x$ is one-to-one

Properties of Injective Functions:

• Composition of injective functions is injective

• If $g \circ f$ is injective, then $f$ is injective (but $g$ may not be)

2.2 Onto (Surjective) Function

A function where every element in the codomain has at least one pre-image.

Formal Definition: $f: A \to B$ is surjective if For every $y \in B$, there exists at least one $x \in A$ such that $f(x) = y$

Key Point: Range = Codomain

Examples:

1. $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 2x + 3$ is onto Proof: For any $y \in \mathbb{R}$, solve $y = 2x + 3$ → $x = \frac{y-3}{2} \in \mathbb{R}$

2. $g: \mathbb{R} \to \mathbb{R}$, $g(x) = x^2$ is NOT onto Reason: Negative numbers (e.g., -1) have no pre-image since $x^2 \ge 0$

3. $h: \mathbb{R} \to [0, \infty)$, $h(x) = x^2$ IS onto Now the codomain matches the range!

Properties of Surjective Functions:

• Composition of surjective functions is surjective

• If $g \circ f$ is surjective, then $g$ is surjective (but $f$ may not be)

2.3 Bijective Function (One-to-One Correspondence)

A function that is both injective and surjective.

Properties:

1. Has an inverse function

2. Establishes a perfect pairing between domain and codomain

3. If $A$ and $B$ are finite sets and $f: A \to B$ is bijective, then $n(A) = n(B)$

Examples:

1. $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 2x + 3$ is bijective

   • One-to-one: Yes (linear function with non-zero slope)

   • Onto: Yes (covers all real numbers)

2. $g: (0, \infty) \to \mathbb{R}$, $g(x) = \ln x$ is bijective

2.4 Constant Function

A function whose output value is the same for every input.

Definition: $f: A \to B$ is constant if there exists $c \in B$ such that $f(x) = c$ for all $x \in A$

Examples:

• $f(x) = 5$ for all $x \in \mathbb{R}$

• $g(x) = \pi$ for all $x \in \mathbb{R}$

Properties:

• Neither injective (unless domain has 1 element)

• Surjective only if codomain = {c}

2.5 Identity Function

A function that returns its input unchanged.

Definition: $I_A: A \to A$ defined by $I_A(x) = x$ for all $x \in A$

Properties:

• Bijective

• $f \circ I_A = f$ and $I_A \circ f = f$ for any $f: A \to A$

Example: $I_\mathbb{R}: \mathbb{R} \to \mathbb{R}$, $I_\mathbb{R}(x) = x$

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3. Special Classes of Functions

3.1 Polynomial Functions

Functions of the form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ where $a_n, a_{n-1}, \ldots, a_0$ are constants, and $a_n \neq 0$

Key Terms:

• Degree: Highest power of x with non-zero coefficient (n)

• Coefficients: $a_n, a_{n-1}, \ldots, a_0$

• Leading coefficient: $a_n$

• Constant term: $a_0$

Types of Polynomial Functions:

1. Constant function: Degree 0, $f(x) = c$

2. Linear function: Degree 1, $f(x) = ax + b$

3. Quadratic function: Degree 2, $f(x) = ax^2 + bx + c$

4. Cubic function: Degree 3, $f(x) = ax^3 + bx^2 + cx + d$

Domain: All real numbers ($\mathbb{R}$)

Examples:

1. $f(x) = 3x^2 - 2x + 1$ (Quadratic, degree 2)

2. $g(x) = 5x^3 - x$ (Cubic, degree 3)

3. $h(x) = 7$ (Constant, degree 0)

3.2 Rational Functions

Functions of the form: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials, and $Q(x) \neq 0$

Domain: All real numbers except where denominator = 0

Examples:

1. $f(x) = \frac{1}{x}$, Domain: $\mathbb{R} - \{0\}$

2. $g(x) = \frac{x^2 - 1}{x - 1}$, Domain: $\mathbb{R} - \{1\}$ Note: $g(x) = \frac{(x-1)(x+1)}{x-1} = x+1$ for $x \neq 1$

3. $h(x) = \frac{x^3 + 2x}{x^2 - 4}$, Domain: $\mathbb{R} - \{-2, 2\}$

3.3 Algebraic Functions

Functions constructed using algebraic operations (addition, subtraction, multiplication, division, and roots).

Examples:

1. $f(x) = \sqrt{x}$ (Square root function) Domain: $[0, \infty)$

2. $g(x) = \sqrt[3]{x}$ (Cube root function) Domain: $\mathbb{R}$ (unlike square root)

3. $h(x) = \sqrt{x^2 - 4}$ Domain: $(-\infty, -2] \cup [2, \infty)$ (since $x^2 - 4 \ge 0$)

3.4 Transcendental Functions

Functions that cannot be expressed as algebraic functions.

a) Exponential Functions

$f(x) = a^x$ where $a > 0$ and $a \neq 1$

Properties:

• Domain: $\mathbb{R}$

• Range: $(0, \infty)$

• If $a > 1$: Increasing function

• If $0 < a < 1$: Decreasing function

• Passes through (0, 1) since $a^0 = 1$

Special Case: Natural exponential function $f(x) = e^x$ where $e \approx 2.71828$

b) Logarithmic Functions

Inverse of exponential functions: $f(x) = \log_a x$

Properties:

• Domain: $(0, \infty)$

• Range: $\mathbb{R}$

• Passes through (1, 0) since $\log_a 1 = 0$

• $\log_a a = 1$

Special Cases:

• Common logarithm: $\log_{10} x$ or $\log x$

• Natural logarithm: $\log_e x$ or $\ln x$

Relationship: $\log_a(a^x) = x$ and $a^{\log_a x} = x$

c) Trigonometric Functions

1. Sine function: $f(x) = \sin x$

   • Domain: $\mathbb{R}$

   • Range: $[-1, 1]$

   • Period: $2\pi$

2. Cosine function: $f(x) = \cos x$

   • Domain: $\mathbb{R}$

   • Range: $[-1, 1]$

   • Period: $2\pi$

3. Tangent function: $f(x) = \tan x = \frac{\sin x}{\cos x}$

   • Domain: $\mathbb{R} - \{\frac{\pi}{2} + n\pi: n \in \mathbb{Z}\}$

   • Range: $\mathbb{R}$

   • Period: $\pi$

d) Inverse Trigonometric Functions

1. Arcsine: $f(x) = \sin^{-1} x$ or $\arcsin x$

   • Domain: $[-1, 1]$

   • Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

2. Arccosine: $f(x) = \cos^{-1} x$ or $\arccos x$

   • Domain: $[-1, 1]$

   • Range: $[0, \pi]$

3. Arctangent: $f(x) = \tan^{-1} x$ or $\arctan x$

   • Domain: $\mathbb{R}$

   • Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$

3.5 Even and Odd Functions

Even Functions

Symmetric about the y-axis: $f(-x) = f(x)$ for all x in domain

Examples:

1. $f(x) = x^2$ Check: $f(-x) = (-x)^2 = x^2 = f(x)$

2. $f(x) = \cos x$ Check: $\cos(-x) = \cos x$

3. $f(x) = |x|$

Odd Functions

Symmetric about the origin: $f(-x) = -f(x)$ for all x in domain

Examples:

1. $f(x) = x^3$ Check: $f(-x) = (-x)^3 = -x^3 = -f(x)$

2. $f(x) = \sin x$ Check: $\sin(-x) = -\sin x$

3. $f(x) = x$

Properties:

• Sum/difference of even functions is even

• Sum/difference of odd functions is odd

• Product of two even or two odd functions is even

• Product of even and odd function is odd

• Every function can be written as sum of even and odd parts: $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$ (even part) + (odd part)

3.6 Periodic Functions

A function is periodic if there exists a positive number T such that: $f(x + T) = f(x)$ for all x in domain

Smallest such T is called the fundamental period.

Examples:

1. $f(x) = \sin x$: Period = $2\pi$

2. $f(x) = \cos x$: Period = $2\pi$

3. $f(x) = \tan x$: Period = $\pi$

4. $f(x) = \sin(2x)$: Period = $\pi$

5. $f(x) = \sin^2 x$: Period = $\pi$ (since $\sin^2 x = \frac{1-\cos 2x}{2}$)

Properties:

• If f has period T, then f(ax+b) has period $\frac{T}{|a|}$

• Sum of periodic functions with commensurate periods is periodic

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4. Operations on Functions

4.1 Algebra of Functions

For functions f and g with common domain D:

a) Sum: $(f + g)(x) = f(x) + g(x)$

Domain: Intersection of domains of f and g

Example: $f(x) = \sqrt{x}$, $g(x) = \sqrt{4-x}$

• Dom(f) = $[0, \infty)$

• Dom(g) = $(-\infty, 4]$

• Dom(f+g) = $[0, \infty) \cap (-\infty, 4] = [0, 4]$

b) Difference: $(f - g)(x) = f(x) - g(x)$

Domain: Intersection of domains of f and g

c) Product: $(f \cdot g)(x) = f(x) \cdot g(x)$

Domain: Intersection of domains of f and g

d) Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$

Domain: Intersection of domains of f and g, excluding where $g(x) = 0$

Example: $f(x) = x^2$, $g(x) = x-1$

• $\left(\frac{f}{g}\right)(x) = \frac{x^2}{x-1}$

• Domain: $\mathbb{R} - \{1\}$

e) Scalar Multiplication: $(cf)(x) = c \cdot f(x)$ where c is constant

4.2 Composition of Functions

Applying one function to the result of another.

Definition: If $f: A \to B$ and $g: B \to C$, then the composition $g \circ f: A \to C$ is defined by: $(g \circ f)(x) = g(f(x))$

Important: Order matters! $g \circ f \neq f \circ g$ in general.

Example: $f(x) = x^2$, $g(x) = x + 1$

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

• $(f \circ g)(x) = f(g(x)) = f(x+1) = (x+1)^2 = x^2 + 2x + 1$ Clearly, $g \circ f \neq f \circ g$

Properties of Composition:

1. Associative: $(h \circ g) \circ f = h \circ (g \circ f)$

2. Identity: $f \circ I = I \circ f = f$

3. Inverse: If f is bijective, then $f^{-1} \circ f = I$ and $f \circ f^{-1} = I$

Domain of Composition:

For $g \circ f$:

1. Start with domain of f

2. Exclude any x for which f(x) is not in domain of g

Example: $f(x) = \sqrt{x}$, $g(x) = \frac{1}{x}$

• Dom(f) = $[0, \infty)$

• Dom(g) = $\mathbb{R} - \{0\}$

• For $g \circ f$: $g(f(x)) = \frac{1}{\sqrt{x}}$

• We need: f(x) ∈ Dom(g) ⇒ $\sqrt{x} \neq 0$ ⇒ $x \neq 0$

• Also: x ∈ Dom(f) ⇒ $x \ge 0$

• Therefore: Dom(g∘f) = $(0, \infty)$

4.3 Inverse Functions

The inverse function "undoes" what the original function did.

Definition: If $f: A \to B$ is bijective, then its inverse $f^{-1}: B \to A$ exists and satisfies:

• $f^{-1}(f(x)) = x$ for all $x \in A$

• $f(f^{-1}(y)) = y$ for all $y \in B$

Finding Inverse Function:

1. Start with $y = f(x)$

2. Solve for x in terms of y

3. Swap x and y to get $y = f^{-1}(x)$

Example: Find inverse of $f(x) = 2x + 3$

1. $y = 2x + 3$

2. Solve: $2x = y - 3$ ⇒ $x = \frac{y-3}{2}$

3. Swap: $y = \frac{x-3}{2}$ So $f^{-1}(x) = \frac{x-3}{2}$

Properties of Inverse Functions:

1. Graphical: Graph of $f^{-1}$ is reflection of graph of f about line $y = x$

2. Domain and Range:

   • Dom($f^{-1}$) = Range(f)

   • Range($f^{-1}$) = Dom(f)

3. Composition: $f^{-1} \circ f = I_A$ and $f \circ f^{-1} = I_B$

4. Inverse of Composition: $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$

Horizontal Line Test Revisited:

A function has an inverse if and only if it is one-to-one (bijective onto its range).

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5. Transformation of Functions

Given base function $y = f(x)$:

5.1 Vertical Transformations

1. Vertical shift up by c: $y = f(x) + c$

2. Vertical shift down by c: $y = f(x) - c$

3. Vertical stretch by factor k (k>1): $y = kf(x)$

4. Vertical compression by factor k (0<k<1): $y = kf(x)$

5. Reflection about x-axis: $y = -f(x)$

5.2 Horizontal Transformations

1. Horizontal shift right by c: $y = f(x - c)$ Note: Opposite direction to intuition!

2. Horizontal shift left by c: $y = f(x + c)$

3. Horizontal stretch by factor k (k>1): $y = f\left(\frac{x}{k}\right)$

4. Horizontal compression by factor k (0<k<1): $y = f\left(\frac{x}{k}\right)$

5. Reflection about y-axis: $y = f(-x)$

5.3 Combined Transformations

General Form: $y = af(b(x - c)) + d$ where:

• a: vertical stretch/compression and reflection (about x-axis if negative)

• b: horizontal stretch/compression and reflection (about y-axis if negative)

• c: horizontal shift

• d: vertical shift

Order of Operations: From inside out:

1. Horizontal shift (c)

2. Horizontal stretch/compression (b)

3. Reflection about y-axis (if b<0)

4. Vertical stretch/compression (a)

5. Reflection about x-axis (if a<0)

6. Vertical shift (d)

Example: Transform $y = \sqrt{x}$ to $y = 2\sqrt{3-x} - 1$

1. Start: $y = \sqrt{x}$

2. Reflection about y-axis: $y = \sqrt{-x}$

3. Horizontal shift right by 3: $y = \sqrt{-(x-3)} = \sqrt{3-x}$

4. Vertical stretch by 2: $y = 2\sqrt{3-x}$

5. Vertical shift down by 1: $y = 2\sqrt{3-x} - 1$

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6. Real-Valued Functions of Real Variables

Functions where both domain and codomain are subsets of $\mathbb{R}$.

6.1 Finding Domain

For real-valued functions, exclude values that make:

1. Denominator zero (for rational functions)

2. Expression under even root negative (for root functions)

3. Argument of logarithm non-positive (for logarithmic functions)

4. Base of exponential negative or 1 (for exponential functions)

Example 1: $f(x) = \frac{\sqrt{x-2}}{x^2 - 9}$ Conditions:

1. For numerator: $x-2 \ge 0$ ⇒ $x \ge 2$

2. For denominator: $x^2 - 9 \neq 0$ ⇒ $x \neq \pm 3$ Combine: $x \ge 2$ and $x \neq 3$ Domain: $[2, 3) \cup (3, \infty)$

Example 2: $g(x) = \ln(4 - x^2)$ Condition: $4 - x^2 > 0$ ⇒ $x^2 < 4$ ⇒ $-2 < x < 2$ Domain: $(-2, 2)$

6.2 Finding Range

Methods:

1. Analytical: Solve $y = f(x)$ for x in terms of y, determine values of y for which solution exists

2. Graphical: Sketch graph, observe y-values

3. Using calculus: Find maxima/minima

Example: Find range of $f(x) = \frac{x}{x^2 + 1}$ Let $y = \frac{x}{x^2 + 1}$ Solve for x: $yx^2 + y = x$ ⇒ $yx^2 - x + y = 0$ For real x, discriminant ≥ 0: $(-1)^2 - 4y^2 \ge 0$ ⇒ $1 - 4y^2 \ge 0$ ⇒ $y^2 \le \frac{1}{4}$ ⇒ $-\frac{1}{2} \le y \le \frac{1}{2}$ Range: $[-\frac{1}{2}, \frac{1}{2}]$

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7. Piecewise Defined Functions

Functions defined by different formulas on different parts of their domain.

General Form:

$$f(x) = \begin{cases} f_1(x) & \text{if } x \in D_1 \\ f_2(x) & \text{if } x \in D_2 \\ \vdots \\ f_n(x) & \text{if } x \in D_n \end{cases}$$

Example 1: Absolute value function

$$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

Example 2: Signum function

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

Example 3: Greatest integer function (Floor function) $f(x) = \lfloor x \rfloor = \text{greatest integer} \le x$ Example: $\lfloor 2.7 \rfloor = 2$, $\lfloor -1.3 \rfloor = -2$

Important: Check continuity at transition points.

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8. Solved Examples

Example 1: Function Composition and Domain

Given $f(x) = \sqrt{x-1}$ and $g(x) = \frac{1}{x^2-4}$, find:

1. $(f \circ g)(x)$ and its domain

2. $(g \circ f)(x)$ and its domain

Solution:

1. For $f \circ g$: $(f \circ g)(x) = f(g(x)) = \sqrt{g(x) - 1} = \sqrt{\frac{1}{x^2-4} - 1}$ Domain conditions:

   • For g: $x^2 - 4 \neq 0$ ⇒ $x \neq \pm 2$

   • For f∘g: $g(x) - 1 \ge 0$ ⇒ $\frac{1}{x^2-4} - 1 \ge 0$ Solve: $\frac{1}{x^2-4} \ge 1$ Case 1: If $x^2-4 > 0$ ($x<-2$ or $x>2$), then $1 \ge x^2-4$ ⇒ $x^2 \le 5$ ⇒ $-\sqrt{5} \le x \le \sqrt{5}$ Intersection: $(2, \sqrt{5}]$

     Case 2: If $x^2-4 < 0$ ($-2<x<2$), then $1 \le x^2-4$ ⇒ $x^2 \ge 5$ ⇒ $x \le -\sqrt{5}$ or $x \ge \sqrt{5}$ Intersection: None with $(-2,2)$

   Final domain for f∘g: $(2, \sqrt{5}]$

2. For $g \circ f$: $(g \circ f)(x) = g(f(x)) = \frac{1}{(f(x))^2 - 4} = \frac{1}{(\sqrt{x-1})^2 - 4} = \frac{1}{x-1-4} = \frac{1}{x-5}$ Domain conditions:

   • For f: $x-1 \ge 0$ ⇒ $x \ge 1$

   • For g∘f: Denominator ≠ 0 ⇒ $x-5 \neq 0$ ⇒ $x \neq 5$ Combine: $[1, 5) \cup (5, \infty)$

Example 2: Finding Inverse Function

Find inverse of $f(x) = \frac{2x+3}{x-1}$ and verify.

Solution:

1. Let $y = \frac{2x+3}{x-1}$

2. Solve for x: $y(x-1) = 2x+3$ $yx - y = 2x + 3$ $yx - 2x = y + 3$ $x(y-2) = y+3$ $x = \frac{y+3}{y-2}$

3. Swap x and y: $y = \frac{x+3}{x-2}$ So $f^{-1}(x) = \frac{x+3}{x-2}$

Verification:

• $f^{-1}(f(x)) = f^{-1}\left(\frac{2x+3}{x-1}\right) = \frac{\frac{2x+3}{x-1}+3}{\frac{2x+3}{x-1}-2} = \frac{2x+3+3(x-1)}{2x+3-2(x-1)} = \frac{5x}{5} = x$

• $f(f^{-1}(x)) = f\left(\frac{x+3}{x-2}\right) = \frac{2\left(\frac{x+3}{x-2}\right)+3}{\frac{x+3}{x-2}-1} = \frac{2x+6+3(x-2)}{x+3-(x-2)} = \frac{5x}{5} = x$

Example 3: Even/Odd Function Analysis

Determine if $f(x) = \frac{e^x + e^{-x}}{2}$ is even, odd, or neither.

Solution: Check $f(-x)$: $f(-x) = \frac{e^{-x} + e^{-(-x)}}{2} = \frac{e^{-x} + e^x}{2} = \frac{e^x + e^{-x}}{2} = f(x)$ Since $f(-x) = f(x)$, the function is even.

Note: This function is actually $\cosh x$, the hyperbolic cosine function.

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9. Practice Tips for Exams

1. Function Notation: Remember $f(x)$ means value at x, not f times x

2. Domain First: Always find domain before other analysis

3. One-to-One Test: Use horizontal line test or algebraic check

4. Inverse Existence: Function must be bijective to have inverse

5. Composition Order: $(g \circ f)(x) = g(f(x))$, apply f first

6. Transformation Order: Inside out for $af(b(x-c))+d$

7. Piecewise Functions: Check endpoints for continuity

8. Even/Odd: Test $f(-x)$ directly

9. Range Finding: Solve $y = f(x)$ for x, find y for real x

10. Graphs: Sketch when possible for visualization

This comprehensive theory covers all aspects of functions with detailed explanations and examples, providing complete preparation for the entrance examination.