Find the domain of the function extended-left-parenthesis Start Fraction f over g End Fraction extended-right-parenthesis left-parenthesis x right-parenthesis where lower f left-parenthesis x right-parenthesis equals x squared minus 9 and lower g left-parenthesis x right-parenthesis equals x squared minus 4 x plus 3.
(1 point)
Responses
left-parenthesis negative infinity comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma infinity right-parenthesis
left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma infinity right-parenthesis
left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma 3 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 negative 1 right-parenthesis union left-parenthesis negative 1 comma infinity right-parenthesis
The domain of the function f/g(x) is the set of all real numbers except for the values of x that make the denominator equal to zero.
To find these values, we need to solve the equation g(x) = x^2 - 4x + 3 = 0. Factoring the quadratic equation, we get (x - 1)(x - 3) = 0. Setting each factor equal to zero, we find that x = 1 and x = 3.
Therefore, the domain of the function f/g(x) is (-∞, 1) ∪ (1, 3) ∪ (3, ∞).
Image with alt text: (-∞, 1) ∪ (1, 3) ∪ (3, ∞)
To find the domain of the function f/g, where f(x) = x^2 - 9 and g(x) = x^2 - 4x + 3, we need to determine the values of x for which g(x) is non-zero.
First, let's find the values of x that make g(x) equal to zero by setting g(x) = 0 and solving for x:
x^2 - 4x + 3 = 0
This equation can be factored as (x - 1)(x - 3) = 0. So, the values of x that make g(x) equal to zero are x = 1 and x = 3.
Therefore, the domain of f/g includes all real numbers except for these values. So, the domain is:
(-∞, 1) ∪ (1, 3) ∪ (3, ∞)