what is the domain of the function?\n$$f(x) = \\frac{x + 6}{(x - 7)(x + 5)}$$\na all real numbers except…

what is the domain of the function?\n$$f(x) = \\frac{x + 6}{(x - 7)(x + 5)}$$\na all real numbers except 7\nb all real numbers except 5 and -7\nc all real numbers except -5 and 7\nd all real numbers

what is the domain of the function?\n$$f(x) = \\frac{x + 6}{(x - 7)(x + 5)}$$\na all real numbers except 7\nb all real numbers except 5 and -7\nc all real numbers except -5 and 7\nd all real numbers

Answer

Explanation:

Step1: Identify the condition for domain

The denominator of a rational function cannot be zero. $$ (x - 7)(x + 5) \neq 0 $$

Step2: Solve for excluded values

Set each factor in the denominator to zero. $$ x - 7 = 0 \implies x = 7 $$ $$ x + 5 = 0 \implies x = -5 $$

Step3: State the domain

The domain includes all real numbers except the roots. $$ x \in \mathbb{R}, x \neq 7, x \neq -5 $$

Answer:

C. all real numbers except -5 and 7