Find the domain of the function f(x) = \frac{x+10}{x^2 + 2x - 35}.

Options:
1. {x | x \ne -7 and x \ne 5}
2. {x | x \ne -35}
3. {x | x \ne 35 and x \ne -10}
4. {x | x \ne 35}
5. {x | x \ne 10}

The domain of the function f ( x ) = x 2 + 2 x − 35 x + 10 ​ is all real numbers except where the denominator equals zero. Specifically, the function is undefined at x = − 7 and x = 5 , so the correct option is 1: { x | x \ne -7 and x \ne 5 }.
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Answered by Anonymous | 2025-07-04

To find the domain of the function f ( x ) = x 2 + 2 x − 35 x + 10 ​ , we need to determine the values of x for which the function is defined.
The function is undefined wherever the denominator is zero. So, we must solve the equation x 2 + 2 x − 35 = 0 .
We'll do this by factoring the quadratic expression:
x 2 + 2 x − 35 can be factored into ( x − 5 ) ( x + 7 ) .
Setting each factor equal to zero gives us the values that make the denominator zero:

x − 5 = 0 gives x = 5
x + 7 = 0 gives x = − 7

Therefore, x = 5 and x = − 7 are the values that would make the denominator zero, and thus, the function undefined at these points.
The domain of the function is all real numbers except − 7 and 5 . We can write this in set notation as:
{ x ∣ x  = − 7 and x  = 5 }
Based on the options given, the correct answer is:

{x \mid x \neq -7 \text{ and } x \neq 5 }

Answered by SophiaElizab | 2025-07-06