find the domain of the following function. State your answer in interval notation. f(x) = 5x + 10/ x + sqrt -26x - 169
To find the domain of a function, we need to identify any values of x that would result in undefined or imaginary outputs. In this case, we need to consider the denominator of the fraction and the expression under the square root.
First, let's focus on the denominator of the fraction, which is (x + sqrt(-26x - 169)). For the function to be defined, this denominator should not equal zero. Thus, we need to solve the equation:
x + sqrt(-26x - 169) ≠ 0
To solve this equation, we need to isolate the square root term:
sqrt(-26x - 169) ≠ -x
Next, we need to square both sides of the equation to eliminate the square root:
-26x - 169 ≠ x^2
Now, let's simplify and rearrange the equation:
x^2 + 26x + 169 ≠ 0
We can factor this quadratic equation as:
(x + 13)^2 ≠ 0
Since (x + 13)^2 is always non-zero for any value of x, we can say that x + sqrt(-26x - 169) is never equal to zero. Therefore, the only restriction on the domain comes from the square root term being defined.
To ensure the square root is defined, we need to make sure that the expression inside the square root is non-negative:
-26x - 169 ≥ 0
Solving this inequality will give us the range of x values for which the function is defined:
-26x ≥ 169
Dividing both sides by -26 (remember to flip the inequality when dividing by a negative number), we get:
x ≤ -169/26
Therefore, the domain of the function f(x) = 5x + 10/ x + sqrt(-26x - 169) in interval notation is:
(-∞, -169/26]