Question

find the domain of the rational function. express your answer in set builder notation.

1. $f(x)=\frac{x+1}{(x+3)(x-4)}$
2. $f(x)=\frac{x-7}{(x+4)(x-9)}$
3. $f(x)=\frac{3x}{(x+1)^2}$
4. $f(x)=\frac{(x-9)}{x^2 - 4x+3}$

1. $\\{x|x \text{ is a real number, } x\

eq -3, x\
eq 4\\}$

1. $\\{x|x \text{ is a real number, } x\

eq -4, x\
eq 9\\}$

1. $\\{x|x \text{ is a real number, } x\

eq -1\\}$

1. $\underline{\quad\quad\quad\quad\quad}$

Explanation:

Step1: Set denominator ≠ 0

$x^2 - 4x + 3
eq 0$

Step2: Factor the quadratic

$(x-1)(x-3)
eq 0$

Step3: Solve for excluded x-values

$x-1
eq 0 \implies x
eq 1$; $x-3
eq 0 \implies x
eq 3$

Step4: Write domain in set-builder

All real numbers except 1 and 3

Answer:

$\{x \mid x \text{ is a real number}, x
eq 1, x
eq 3\}$