# 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}$$ passes vertical line test → Function *** ### **2. Types of Functions** #### **2.1 One-to-One (Injective) Function** A function where **different inputs give different outputs**. **Formal Definition:** $$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$$ *** ### **3. Special Classes of Functions** #### **3.1 Polynomial Functions** Functions of the form: $$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|}$$ * Sum of periodic functions with commensurate periods is periodic *** ### **4. Operations on Functions** #### **4.1 Algebra of Functions** For functions f and g with common domain D: **a) Sum:** $$(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}$$ **Horizontal Line Test Revisited:** A function has an inverse if and only if it is one-to-one (bijective onto its range). *** ### **5. Transformation of Functions** Given base function $$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)$$ 4. **Vertical compression by factor k (0\1):** $$y = f\left(\frac{x}{k}\right)$$ 4. **Horizontal compression by factor k (0\ 0$$ ⇒ $$x^2 < 4$$ ⇒ $$-2 < x < 2$$ 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}]$$ *** ### **7. Piecewise Defined Functions** Functions defined by different formulas on different parts of their domain. **General Form:** $$ 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$$ **Important:** Check continuity at transition points. *** ### **8. Solved Examples** #### **Example 1:** Function Composition and Domain Given $$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\