What is the domain (in interval notation) of the following functions?

1. \( g(x) = \frac{3}{5x-4} \)

2. \( h(x) = \frac{\sqrt{x}}{x-5} \)

3. \( f(x) = \frac{\sqrt{x}}{x^2-5x} \)

4. \( g(x) = \frac{\sqrt{x}+5}{x^2-x-20} \)

5. \( h(x) = \frac{3}{x^2+1} \)

6. \( f(x) = \frac{\sqrt{x-2}}{x+1} \)

7. \( g(x) = \frac{x^2}{3x^2-x-2} \)

8. \( h(x) = 3(x-4)^2-7 \)

Note: Number sets in parenthesis are either on top of or beneath the fraction bar, and ^2 here represents a number squared.

1. g ( x ) = 5 x − 4 3 ​ D : 5 x − 4  = 0 → 5 x  = 4 / : 5 → x  = 5 4 ​ → x ∈ R \ { 5 4 ​ } 2. h ( x ) = x − 5 x ​ ​ D : x ≥ 0 ∧ x − 5  = 0 → x ≥ 0 ∧ x  = 5 → x ∈ ⟨ 0 ; ∞ ) \ { 5 }
3. f ( x ) = x 2 − 5 x x ​ ​ D : x ≥ 0 ∧ x 2 − 5 x  = 0 → x ≥ 0 ∧ x ( x − 5 )  = 0 → x ≥ 0 ∧ x  = 0 ∧ x  = 5 → x ∈ R + \ { 5 } 4. g ( x ) = x 2 − x − 20 x ​ + 5 ​ D : x ≥ 0 ∧ x 2 − x − 20  = 0 → x ≥ 0 ∧ ( x + 4 ) ( x − 5 )  = 0 → x ≥ 0 ∧ x  = − 4 ∧ x  = 5 → x ∈ ⟨ 0 ; ∞ ) \ { − 4 ; 5 }
5. h ( x ) = x 2 + 1 3 ​ D : x 2 + 1  = 0 → x 2  = − 1 → x ∈ R 6. f ( x ) = x + 1 x − 2 ​ ​ D : x − 2 ≥ 0 ∧ x + 1  = 0 → x ≥ 2 ∧ x  = − 1 → x ∈ ⟨ 2 ; ∞ )
7. g ( x ) = 3 x 2 − x − 2 x 2 ​ D : 3 x 2 − x − 2  = 0 → ( 3 x + 2 ) ( x − 1 )  = 0 → x  = − 3 2 ​ ∧ x  = 1 → x ∈ R \ { − 3 2 ​ ; 1 } 8. h ( x ) = 3 ( x − 4 ) 2 − 7 D : x ∈ R

Answered by Anonymous | 2024-06-10

The domains for the functions are identified by avoiding undefined situations like division by zero or negative square roots. Each function has been analyzed step-by-step to determine its respective domain in interval notation. Domains vary from all real numbers to specific intervals excluding certain values.
;

Answered by Anonymous | 2024-12-24