MTH100 Section Incharge Arfan Ali by Ali Khan

MTH-100 · Section In charge : Muhammad Arfan Ali · Vu nexus · Ali Khan

Q1. Let F (x) =

1. Determine The domain and range of each functions :

Domain: To find the domain of f(x) we need to ensure that the denominator x2-3 is not zero, as division by zero is undefined.

1. Set the denominator equal to zero: X2-3 X2= 3 X = ±√3

2. So, f(x) is undefined at x= √3 and x= - √3

Domain:

(-∞ , -√3) U (-√3 ,√3) U ( √3 , ∞)

Range:

Since this is a rational function with a cubic polynomial in the numerator and a quadratic polynomial in the denominator, it will approach positive or negative infinity as x→−∞ and x→- ∞ However, determining the exact range analytically is complex and typically requires calculus (finding asymptotes, critical points, etc.). Therefore, we can say:

Range: (-∞, ∞) except possibly at values excluded by vertical asymptotes or points of discontinuity.

2. Function : G(x) = √19 −𝑥2

Domain:

For g(x) to be defined, the expression inside the square root must be non-negative:

1. Set the inside of the square root to be greater than or equal to zero: 19-x2 ≥0 X2 ≤19 -√19 ≤𝑥 ≤√19 2. So, g(x) is defined for x in the interval [−√19 , √19]

Domain: [−√19 √19]

Range: Since g(x) = √19 − 𝑥2 represents the upper half of a circle with radius√19, the values of g(x) range from 0 to √19

Range: (0,√𝟏𝟗)

Summary

For f(x) = 𝑥2−3

Domain: (-∞,−√3 ) U (-√3,√3) U (√3, ∞) Range: (-∞, ∞) (except possibly at points of discontinuity)

For g(x) = √19 − 𝑥2

Domain: (−√19,√19) Range: (0,√19)

b) Classify each function as even odd or neither justify your answer with the necessary algebraic Checks

To determine whether each function is even, odd, or neither, we need to apply the definitions of even and odd functions.

A function f(x)is even if f(-x)= -f(x) for all x in its domain. A function f(x) is odd if f(-x) = f(x) for all in its domain. If neither of these conditions is satisfied, the function is classified as neither even nor odd.

Let’s check each function.

1. Function F(x)

To determine if f(x) is even, odd, or neither, let’s calculate f(-x)

1. Substitute -x for x in f(x) : F(-x) = (−𝑥)2−3 2. Simplify f(-x) F(-x) = 𝑥2−3