Find the domain of the function [tex]f(x) = \frac{x+6}{x^2 + 10x - 11}[/tex].

Options:
1. [tex]{x | x \ne -11}[/tex] and [tex]x \ne 1[/tex]
2. [tex]{x | x \ne -11}[/tex]
3. [tex]{x | x \ne 11}[/tex]
4. [tex]{x | x \ne 6}[/tex]
5. [tex]{x | x \ne 11}[/tex] and [tex]x \ne -6}[/tex]

The domain of the function f ( x ) = x 2 + 10 x − 11 x + 6 ​ excludes the values − 11 and 1 . Therefore, the correct answer is option 1: { x ∣ x  = − 11 , x  = 1 } . This means you can use all real numbers except for those two values.
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Answered by Anonymous | 2025-07-04

To find the domain of the function f ( x ) = x 2 + 10 x − 11 x + 6 ​ , we need to determine the set of all possible values of x for which the function is defined.
The function is undefined where the denominator is zero because division by zero is not allowed. Therefore, we first find the values of x that make the denominator zero.
The denominator is x 2 + 10 x − 11 . To find the values that make this zero, set it equal to zero and solve for x :
x 2 + 10 x − 11 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = 2 a − b ± b 2 − 4 a c ​ ​
where a = 1 , b = 10 , and c = − 11 . Plug these values into the quadratic formula:
x = 2 ⋅ 1 − 10 ± 1 0 2 − 4 ⋅ 1 ⋅ ( − 11 ) ​ ​
x = 2 − 10 ± 100 + 44 ​ ​
x = 2 − 10 ± 144 ​ ​
x = 2 − 10 ± 12 ​
This gives us two solutions:

x = 2 − 10 + 12 ​ = 1

x = 2 − 10 − 12 ​ = − 11


Therefore, the values of x that make the denominator zero are x = 1 and x = − 11 .
The domain of the function is all real numbers except x = 1 and x = − 11 . In set-builder notation, the domain is:
{ x ∣ x  = − 11 and x  = 1 }
Thus, the correct multiple choice option is 1. { x ∣ x  = − 11 and x  = 1 } .

Answered by BenjaminOwenLewis | 2025-07-06