Question
Find the domain of the function . Write the answer in interval notation.
lower f left parenthesis x right parenthesis equals Start Fraction 2 x superscript 3 baseline minus 250 over x superscript 2 baseline minus 2 x minus 15 End Fraction
(1 point)
Responses
left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis negative 3 comma 5 right parenthesis union left parenthesis 5 comma infinity right parenthesis
Image with alt text: left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis negative 3 comma 5 right parenthesis union left parenthesis 5 comma infinity right parenthesis
left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis
Image with alt text: left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis negative 5 comma 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis
left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis 3 comma infinity right parenthesis
Image with alt text: left parenthesis negative infinity comma negative 5 right parenthesis union left parenthesis 3 comma infinity right parenthesis
left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis negative 3 comma infinity right parenthesis
The domain of the function is all the values of x for which the function is defined. In this case, the only x-values that are not allowed are those that make the denominator zero, since division by zero is undefined.
To find these values, we set the denominator equal to zero and solve for x:
x^2 - 2x - 15 = 0
This can be factored as (x - 5)(x + 3) = 0
Setting each factor equal to zero, we get x - 5 = 0 and x + 3 = 0
Solving these equations, we find x = 5 and x = -3
Since the function is not defined for x = 5 and x = -3, the domain is all real numbers except x = 5 and x = -3.
Therefore, the domain of the function is (-∞, -3) ∪ (-3, 5) ∪ (5, ∞), or in interval notation: (-∞, -3) ∪ (-3, 5) ∪ (5, ∞)
To find the domain of the function, we need to determine the x-values for which the function is defined.
The function is defined for all real numbers except those that would result in a division by zero or a square root of a negative number.
To find these x-values, we set the denominator of the fraction equal to zero and solve for x.
x^2 - 2x - 15 = 0
Factorizing the quadratic equation, we get:
(x-5)(x+3) = 0
Setting each factor equal to zero, we find:
x-5 = 0 or x+3 = 0
x = 5 or x = -3
Therefore, the function is undefined when x = 5 or x = -3.
So, the domain of the function is:
(-∞, -3) ∪ (-3, 5) ∪ (5, ∞)
In interval notation, this is represented as:
(-∞, -3) ∪ (-3, 5) ∪ (5, ∞)