Pre Calculus Unit 3.pptx Functions By Dr.Mamoona Anam

✅ What Are Functions?
A function is a special type of relation that connects each input to exactly one output.

In simple terms:

A function assigns only one output for each input.

📌 Definition:
A function
𝑓
f from set
𝐴
A to set
𝐵
B is a rule that assigns each element in
𝐴
A exactly one element in
𝐵
B.

We write:

𝑓
:
𝐴
→
𝐵
f:A→B
If
𝑥
∈
𝐴
x∈A, then the output is written as
𝑓
(
𝑥
)
∈
𝐵
f(x)∈B.

📈 Example:
Let
𝑓
(
𝑥
)
=
𝑥
2
f(x)=x
2


If
𝑥
=
2
x=2, then
𝑓
(
2
)
=
4
f(2)=4

If
𝑥
=
−
3
x=−3, then
𝑓
(
−
3
)
=
9
f(−3)=9

This means the function takes a number and squares it.

🔢 Examples of Functions:
𝑓
(
𝑥
)
=
𝑥
+
1
f(x)=x+1

𝑓
(
𝑥
)
=
2
𝑥
2
−
3
𝑥
+
1
f(x)=2x
2
−3x+1

𝑓
(
𝑥
)
=
𝑥
f(x)=
x
​


𝑓
(
𝑥
)
=
∣
𝑥
∣
f(x)=∣x∣

🧮 Not a Function Example:
𝑓
(
𝑥
)
=
±
𝑥
f(x)=±
x
​

This is not a function because one input (like
𝑥
=
4
x=4) gives two outputs:
+
2
+2 and
−
2
−2. A function must give only one output per input.

📚 Important Terms:
Domain: The set of all possible inputs (x-values)

Range: The set of all possible outputs (f(x)-values)

Co-domain: The set that contains the range; it's where the outputs live

🧠 In Summary:
A function is a rule that links every input to one and only one output.

1. Pre Calculus Unit 3 By Dr.Mamoona Anam Functions By Dr.Mamoona Anam

2. Functions Functions By Dr.Mamoona Anam

3. Introduction to Functions In this unit, the notion of functions is introduced. To start with, methods to compute domain and range are also discussed. The use of arrow diagram for functions is also discussed. In the latter half of the unit, the types of functions are introduced along with the associated examples. These type include trigonometric, exponential, even and odd functions. Functions By Dr.Mamoona Anam

4. What are Functions Functions are considered as the basic tools to understand calculus. Basically functions are used to explain relationships between two or more variable quantities and due to this relation some quantities become independent and others are called dependent quantities. The word “function” was introduced by Leibniz who is considered as the founder of Calculus along with Newton in the 18 th Century. We often see that in real world most of quantities or variables are dependent on each other. For example, the price of wheat depends upon the yield produced in a certain year, your success in exam depends upon the time given to studies during the semester, the volume of a cylinder depends upon its height and radius of its circular base. Functions By Dr.Mamoona Anam

5. Functions Functions are considered as relations between a set of input quantities and a set of output quantities with certain properties. In other words, functions can be imagined as a machine which has two ends. We put some quantities into this machine from one end and these input quantities or variables some how get changed inside the machine to deliver us a new product on the other end which is called output or product of this machine. As we said earlier that functions are to be considered as relation. So we need to understand and explain relations first to understand the theory related to functions. Functions By Dr.Mamoona Anam

6. Relationship of Variables The relation can be between two quantities or variables of the same nature or of different natures. For example, consider the monthly pay of a salesman. He draws a basic pay of Rs. 40,000 and further he is awarded a commission of Rs. 500 for every unit of a good or variable he sells out during the whole month. So in this way we observe that his monthly pay involves two quantities. The first quantity is his basic pay which is fixed and the second variable is the number of goods he has sold out in a particular month. Functions By Dr.Mamoona Anam

7. Mathematically Proof of Function Let us denote his monthly pay by y , and the second variable (no. of goods he sold) by x , then by using this information we can write this monthly pay mathematically as follows: y = 40 , 000 + 500 x This repeats a relation between x and y , e.g., if he has sold no good during the month i.e., x = 0 then he gets only basic pay of Rs. 40,000. This can be represented in the relation form as ( x, y ) = (0 , 40 , 000) Functions By Dr.Mamoona Anam

8. Objectives Functions By Dr.Mamoona Anam

9. functions and its domain and range. Functions By Dr.Mamoona Anam

10. functions and its domain and range. Functions By Dr.Mamoona Anam

11. functions and its domain and range. Functions By Dr.Mamoona Anam

12. functions and its domain and range. Functions By Dr.Mamoona Anam

13. Independent and dependent variables of a function. A function relates an input (independent variable) to exactly one output (dependent variable). Independent Variable It is the input of the function. Usually represented by x You can choose or control its value. Dependent Variable It is the output of the function. Usually represented by y or f(x) Its value depends on the input . Example In the function y=2x+3y x is the independent variable . y is the dependent variable , because its value changes based on x. Functions By Dr.Mamoona Anam

14. What is an Arrow Diagram? An arrow diagram is a visual tool used to show how elements from a domain (input set) map to elements in a range (output set). It uses arrows to connect each input to its corresponding output, helping you see whether the relation is a function and what type of function it is. Functions By Dr.Mamoona Anam

15. ✅ 1. One-to-One Function (Injective) ➤ Definition: Each element in the domain maps to a unique element in the range. No two different inputs have the same output. Domain: A B C ↓ ↓ ↓ Range: 1 2 3 ➤ Explanation: All inputs have different outputs. Each output is used once . This is a valid function and also an injective one. Functions By Dr.Mamoona Anam

16. ✅ 2. Many-to-One Function ➤ Definition: Two or more elements in the domain map to the same element in the range. Domain: A B C ↓ ↓ ↓ Range: 1 1 2 Explanation: Input A and B both map to 1. It is still a function , because each input has only one output . This is not one-to-one , but it is a valid function. Functions By Dr.Mamoona Anam

17. ❌ 3. Not a Function ➤ Definition: An element in the domain maps to more than one element in the range. This violates the definition of a function. Domain: A ↙ ↘ Range: 1 2 ➤ Explanation: One input (A) gives multiple outputs , which breaks the rule. This is not a function . Functions By Dr.Mamoona Anam

18. ✅ 4. Onto Function (Surjective) ➤ Definition: Every element in the range is covered — there is at least one arrow pointing to every element in the range. Domain: A B C ↓ ↓ ↓ Range: 1 2 3 ➤ Explanation: All outputs are used. It doesn’t matter if two inputs go to the same output, as long as all outputs are covered. This is a surjective function . Functions By Dr.Mamoona Anam

19. ✅ Summary Table Function Type Rule Is It a Function? One-to-One Each input → unique output ✅ Yes Many-to-One Multiple inputs → same output ✅ Yes Not a Function One input → more than one output ❌ No Onto (Surjective) Every output has at least one arrow from input ✅ Yes Functions By Dr.Mamoona Anam

20. Types of Functions A. Based on Input-Output Relationship (Mapping Types) B. Based on Equation Type (Algebraic Functions) C. Other Important Types D. Special Functions Functions By Dr.Mamoona Anam

21. 🔷 A. Based on Input-Output Relationship (Mapping Types) Functions By Dr.Mamoona Anam

22. 1. One-to-One Function (Injective) ✅ Definition: one-to-one function , also called an injective function , is a function where each element of the domain maps to a unique element in the codomain . In simpler terms: No two different inputs have the same output. 📉 Graphical Representation: A one-to-one function passes the Horizontal Line Test : Any horizontal line intersects the graph at most once . This ensures that no y-value is repeated . 🧠 Real-Life Analogy: Imagine assigning each student in a class a unique ID number . No two students share the same ID → This is a one-to-one assignment. Functions By Dr.Mamoona Anam

23. Common one to one Function Function Type Example One-to-One? Linear (non-zero slope) f(x)=5x−7f(x) = 5x - 7 ✅ Yes Cubic f(x)=x3f(x) = x^3 ✅ Yes Exponential f(x)=2xf(x) = 2^x ✅ Yes Quadratic f(x)=x2f(x) = x^2 ❌ No Functions By Dr.Mamoona Anam

24. Many-to-One Function ✅ Definition: many-to-one function is a function where two or more different inputs in the domain map to the same output in the codomain . In other words: A single output value corresponds to multiple input values . 📉 Graphical Representation: Fails the Horizontal Line Test : A horizontal line intersects the graph more than once , showing repeated y-values. Graph of f(x)=x2f(x) = x^2f(x)=x2 : A U-shaped parabola where many x-values (e.g., ±2,±3) give the same y-value. 🧠 Real-Life Analogy: Think of calculating age from birth year : A person born in 2000 and another in 2001 might both be 24 years old in 2025. Different birth years (inputs) → Same age (output) → Many-to-One Functions By Dr.Mamoona Anam

25. Common many to one function Function Type Example Many-to-One? Quadratic f(x)=x2f(x) = x^2 ✅ Yes Cosine function f(x)= cos⁡xf (x) = cos x ✅ Yes Absolute value ( f(x) = x Identity function f(x)=xf(x) = x ❌ No Functions By Dr.Mamoona Anam

26. 3. Onto Function (Surjective) ✅ Definition: onto function , also called a surjective function , is a function where every element in the codomain has at least one pre-image in the domain. In simple terms: Every element of the codomain is the output of at least one input . 📉 Graphical Representation: The function's graph covers the entire codomain vertically . There are no "unused" output values . 🧠 Real-Life Analogy: Imagine a delivery service that drops off packages to every house in a neighborhood. Every house (codomain element) gets a delivery (output). There are no skipped houses → That's onto . Functions By Dr.Mamoona Anam

27. Examples of onto function Function Onto over? Surjective? f(x)=x3 R→R ✅ Yes f(x)=sin(x) R→[−1,1] ✅ Yes f(x)=x^2 R→R ❌ No f(x)= 2x + 1 Z→Z ✅ Yes Functions By Dr.Mamoona Anam

28. One-to-One and Onto (Bijective) 3. Bijective Function: A function is called Bijective if it is both One-to-One (Injective) and Onto (Surjective). In a bijective function, each element in the domain maps to a unique element in the codomain, and every element in the codomain is covered by the mapping. This means there is a perfect "pairing" between the domain and codomain. Functions By Dr.Mamoona Anam

29. Summary Injective (One-to-One) : No two different elements in the domain map to the same element in the codomain. Surjective (Onto) : Every element in the codomain is mapped to by at least one element in the domain. Bijective : The function is both injective and surjective, meaning there is a perfect one-to-one correspondence between the domain and the codomain. Functions By Dr.Mamoona Anam

30. 🔷 B. Based on Equation Type (Algebraic Functions) 1. Linear Function 2. Quadratic Function 3. Cubic Function 4. Polynomial Function 5. Rational Function Functions By Dr.Mamoona Anam

31. 1. Linear Function A linear function is a type of function that creates a straight line when graphed on the coordinate plane. It has a constant rate of change (slope), and its general form is: f(x)= mx+b Where: f(x) or y is the output (dependent variable), x is the input (independent variable), M is the slope of the line (rate of change), b is the y-intercept , the value of y when x=0 Functions By Dr.Mamoona Anam

32. Key Properties of Linear Functions: 1. Graph is a Straight Line Always forms a straight line when plotted. Direction of the line depends on the slope mmm: m>0m > 0m>0: line slopes upward (increasing function), m<0m < 0m<0: line slopes downward (decreasing function), m=0m = 0m=0: horizontal line (constant function). Functions By Dr.Mamoona Anam

33. 2. Constant Rate of Change The change in y is constant for every unit change in x. This constant change is called the slope , calculated as: ​​ Functions By Dr.Mamoona Anam

34. 3. Domain and Range For most linear functions, both domain and range are all real numbers: Functions By Dr.Mamoona Anam

35. 4. One-to-One (Injective)? Functions By Dr.Mamoona Anam

36. 5. Onto (Surjective)? Functions By Dr.Mamoona Anam

37. 6. Bijective? Functions By Dr.Mamoona Anam

38. 🔷 C. Other Important Types 1. Absolute Value Function 2. Square Root Function 3. Exponential Function 4. Logarithmic Function 5. Trigonometric Functions Functions By Dr.Mamoona Anam

39. 1. Absolute Value Function ✅ What is an Absolute Value Function? An absolute value function is a function that involves the absolute value of a variable. The absolute value of a number is its distance from zero on the number line, regardless of direction — so it is always non-negative . The most common form of the absolute value function is: f(x)=∣x∣ This means: If x≥0 then f(x)=x If x<0 then f(x)=−x So it "flips" negative values to positive and leaves positive values unchanged. Functions By Dr.Mamoona Anam

40. 1. Absolute Value Function 🔍 Graph of f(x)=∣x∣: The graph is V-shaped . The vertex (corner point) is at (0,0). It is symmetric about the y-axis . It consists of two linear pieces: Left of zero: f(x)=−x Right of zero: f(x)=x Functions By Dr.Mamoona Anam

41. 💡 Why is the Absolute Value Function Used? To Represent Distance: Distance is always positive (or zero). For example, the distance between positions x=−5x = -5x= ∣0−(−5)∣=5 To Handle Positive and Negative Inputs Equally: When you want the same output for inputs like −3-3−3 and 333, absolute value is useful: ∣−3∣=∣3∣=3|-3| = |3| = 3∣−3∣=∣3∣=3. In Real-Life Scenarios: Physics: Displacement or speed can be modeled using absolute values. Engineering: Measuring errors or deviations from a set point. Business/Economics: Evaluating profit/loss changes regardless of direction. Piecewise Definitions: Helps define functions that behave differently on either side of a number (like 0). Functions By Dr.Mamoona Anam

42. Summary Feature Absolute Value Function Basic form ( f(x) = Graph shape V-shape Domain R (all real numbers) Range [0,∞) Used for Distance, symmetry, error analysis Functions By Dr.Mamoona Anam

43. 2. Square Root Function ✅ What is a Square Root Function? A square root function is a type of radical function that involves the square root of the input variable. Its most basic form is: f(x)= xf (x) = \sqrt{x}f(x)=x​ This means that for every non-negative xxx, the function gives the positive square root of xxx. For example: f(4)=4=2f(4) = \sqrt{4} = 2f(4)=4​=2 f(9)=9=3f(9) = \sqrt{9} = 3f(9)=9​=3 Functions By Dr.Mamoona Anam

44. 📈 Graph of f(x)=x: The graph starts at (0, 0) and gradually increases , curving upward. It is not defined for negative values of x in the real number system. It grows slower than linear or quadratic functions. Functions By Dr.Mamoona Anam

45. Key Properties Property Description Domain x≥0 (because you can’t take square root of negative numbers in real numbers) Range y≥0 (since square roots are non-negative) Intercept Passes through the origin: (0,0) Increasing? Yes, it increases slowly as x increases Continuity Continuous for all x≥0 Functions By Dr.Mamoona Anam

46. 📦 Why Is the Square Root Function Used? In Geometry: Calculating length of a side using the Pythagorean theorem . Finding diagonal of rectangles or squares: a2+b2\sqrt In Physics and Engineering: Calculating speed , acceleration , and energy . Example: time in free fall is proportional to h\sqrt{h}h​, where h is height. In Statistics: Standard deviation involves square roots: variance\sqrt{\text{variance}}variance​. In Real-Life Modeling: Used when growth slows over time (e.g., certain types of economic models or learning curves). Functions By Dr.Mamoona Anam

47. 🧠 Example: f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2​ The graph shifts 2 units to the right . The domain becomes x≥2x \ geq 2x≥2. Starts from point (2,0)(2, 0)(2,0). Functions By Dr.Mamoona Anam

48. Summary Function Type Square Root Function Standard Form f(x)=xf(x) = \sqrt{x} Graph Shape Gently increasing curve Domain x≥0 Range y≥0 Common Application Geometry, physics, statistics Functions By Dr.Mamoona Anam

49. 4. Logarithmic Function Functions By Dr.Mamoona Anam

50. Key Concepts Functions By Dr.Mamoona Anam

51. Key Concepts Functions By Dr.Mamoona Anam

52. 5. Trigonometric Functions Trigonometric functions relate angles of a triangle to the ratios of its sides. They are fundamental in mathematics, especially in geometry, physics, and engineering. 📐 The 6 Basic Trigonometric Functions Based on a right triangle or the unit circle, here are the six primary trigonometric functions: Function Name Definition (in right triangle) sin⁡ θ Sine Opposite/Hypotenuse cos⁡ θ Cosine Adjacent/Hypotenuse tan⁡ θ Tangent Opposite/Adjacent csc⁡ θ Cosecant (reciprocal of sine) Hypotenuse/Opposite sec⁡ θ Secant (reciprocal of cosine) Hypotenuse/Adjacent cot⁡ θ Cotangent (reciprocal of tangent) Adjacent/Opposite Functions By Dr.Mamoona Anam

53. 📈 Graphs of Trigonometric Functions Functions By Dr.Mamoona Anam

54. 🔷 D. Special Functions Functions By Dr.Mamoona Anam

55. 1. Identity Function ✅ 1. Identity Function The identity function is one of the simplest and most fundamental functions in mathematics. Definition: f(x)=x This means that the output is always equal to the input. Properties: Linear function Odd function : f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x) Continuous and increasing One-to-one and onto Functions By Dr.Mamoona Anam

56. 2. Signum Function ✅ Signum Function (Sign Function) The signum function , often written as sgn (x) , tells you whether a number is positive, negative, or zero . It "extracts" the sign of a real number. Graph: A piecewise function with: A constant value of −1 for negative x-values A value of at x = 0 A constant value of 1 for positive x-values 🟦 Graph has: A jump discontinuity at x=0x = 0x=0 Looks like three horizontal lines Key Properties: Odd function : sgn (−x)=− sgn (x)\text{ sgn }(-x) = -\text{ sgn }(x) sgn (−x)=− sgn (x) Not continuous at x=0x = 0x=0 Often used in piecewise definitions or modulus simplification Functions By Dr.Mamoona Anam

57. 3. Greatest Integer (Floor) Function ✅ Greatest Integer Function (Floor Function) The Greatest Integer Function , also known as the floor function , is written as: f(x)=⌊ x⌋It returns the greatest integer less than or equal to a given real number. 📈 Graph Features: A step function Each step is closed on the left , open on the right Discontinuous at all integers ❗ Key Points: ⌊x⌋≤x<⌊x⌋+1The graph is a series of horizontal segments It jumps at every integer value (discontinuity) Functions By Dr.Mamoona Anam

58. 4. Piecewise Function ✅ 4. Piecewise Function A piecewise function is a function that is defined by different expressions for different parts of its domain . Each "piece" of the function applies to a specific interval of the input values. 📈 Graph Features: May have breaks , jumps , or sharp corners Can be continuous or discontinuous Each piece is graphed only on its specified interval 💡 Uses: Modeling tax brackets Shipping costs Physics (e.g., motion with different velocities over time) Functions By Dr.Mamoona Anam

59. Types of functions Type Key Feature Example One-to-One Unique output for each input f(x)=2x+3f(x) = 2x + 3 Many-to-One Multiple inputs → one output f(x)=x2f(x) = x^2 Onto All outputs covered f(x)=x3f(x) = x^3 Constant Same output for all inputs f(x)=5f(x) = 5 Polynomial Powers of x f(x)=x2−4x+3f(x) = x^2 - 4x + 3 Rational Fraction of polynomials f(x)=1xf(x) = \frac{1}{x} Exponential Variable in exponent f(x)=2xf(x) = 2^x Logarithmic Inverse of exponential f(x)=log⁡xf(x) = \log x Trigonometric Periodic, angles f(x)=sin⁡xf(x) = \sin x Piecewise Multiple rules Different for x<0x < 0 Absolute Value Distance from 0 ( f(x) = Functions By Dr.Mamoona Anam