01_Ex 1.1

Exercise Questions

Question	Link
Q1. Given that (a) $f(x)=x^{2}-x$, Find…	1.1 Q-1
Q2. Find $\frac{f(a+h)-f(a)}{h}$ and simplify where…	1.1 Q-2
Q3. Express the following…	1.1 Q-3
Q4. Find the domain and range of the…	1.1 Q-4
Q5. Given $f(x)=x^{3}-a x^{2}+b x+1$…	1.1 Q-5
Q6. A stone falls from a height of 60 m…	1.1 Q-6
Q7. Show that the parametric equations…	1.1 Q-7
Q8. Prove the identities…	1.1 Q-8
Q9. Determine whether the given function $\boldsymbol{f}$ is even or odd.	1.1 Q-9

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Overview

This Exercise introduces the concept of functions and limits, foundational topics in mathematics with applications across various fields such as calculus, physics, and engineering.

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Key Concepts

Function

A function is a rule that relates each element of a set to a corresponding element of another set.

• Definition:
  $f: X \rightarrow Y$
  Where $X$ is the domain, and $Y$ is the range.

Domain of a Function

The set $X$ is called the domain of the function $f$.

Range of a Function

The set $Y$ is called the range of the function.

Types of Functions

1. Algebraic Functions

   • Polynomial Function:
     $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$
     Coefficients $a_n, a_{n-1}, \dots, a_0$ are real numbers, and exponents are positive integers.
   • Linear Function: A polynomial of degree 1, e.g., $f(x) = 9x + 7$.
   • Identity Function: $f(x) = x$ when $a = 0$ and $b = 0$.
   • Constant Function: $f(x) = C$, where $C$ is constant.
   • Rational Function: $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, $Q(x) \neq 0$.

2. Trigonometric Functions
   Examples include:

   • $y = \cos x$ (Domain $= R$, Range $= [-1, 1]$)
   • $y = \sin x$ (Domain $= R$, Range $= [-1, 1]$)
   • $y = \tan x$ (Domain excludes $(2n+1)\frac{\pi}{2}$, Range $= R$).

3. Inverse Trigonometric Functions
   Examples include:

   • $y = \sin^{-1} x \Leftrightarrow x = \sin y$, Domain $= [-1, 1]$, Range $= \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

4. Exponential Functions
   Functions where the variable is in the exponent, e.g., $y = e^{ax}$, $y = 2^x$.

5. Logarithmic Functions
   Examples include $\log_a x$ and $\ln x$ for $a > 0$, $x > 0$.

6. Explicit Functions
   Functions where $y$ is expressed in terms of $x$, e.g., $y = 2x + 4$.

7. Implicit Functions
   Functions where $x$ and $y$ are mixed, e.g., $x^3 + y^3 + 7xy^2 = 0$.

8. Parametric Functions
   Represented by a third variable, e.g., $x = f(t)$, $y = g(t)$.

9. Even Functions
   $f(-x) = f(x)$, e.g., $f(x) = x^2$.

10. Odd Functions
    $f(-x) = -f(x)$, e.g., $f(x) = x^3$.

Limits and Continuity

• Limit of a Function:
  $\lim_{x \to a} f(x) = L$
  $L$ is the value $f(x)$ approaches as $x$ approaches $a$.
• Continuous Function: A function $f(x)$ is continuous if $\lim_{x \to c} f(x) = f(c)$.
• Discontinuous Function: A function not continuous at $c$.

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Summary

This Exercise covers the fundamental concepts of functions, their types, limits, and continuity, which are essential for understanding advanced topics in mathematics.

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Reference

By Waqas Nasir:

By Sir Shazad Sair: