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

eq 3$

1. ________

Explanation:

Step1: Identify denominator roots

Set denominator equal to 0:
$(x+3)(x-4)=0$
Solve for $x$: $x=-3$ or $x=4$

Step2: Exclude roots from domain

Rational functions are defined where denominator ≠0.
Domain for $f(x)=\frac{x+1}{(x+3)(x-4)}$: all real numbers except $x=-3$ and $x=4$.

Step3: Identify denominator roots (Q4)

Set denominator equal to 0:
$(x+4)(x-9)=0$
Solve for $x$: $x=-4$ or $x=9$

Step4: Exclude roots from domain (Q4)

Rational functions are defined where denominator ≠0.
Domain for $f(x)=\frac{x-7}{(x+4)(x-9)}$: all real numbers except $x=-4$ and $x=9$.

Answer:

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

eq -3, x
eq 4\}$

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

eq -4, x
eq 9\}$