Question

Let f (x) = x4 – 4x3 – 2x2 + 12x + 9, g of x is equal to the square root of the quantity x squared minus 2 times x minus 3 end quantity and h of x is equal to the quantity negative x squared plus 1 end quantity over the quantity x squared minus 2 times x minus 3 end quantity

Part A: Use complete sentences to compare the domain and range of the polynomial function f (x) to that of the radical function g(x).

Part B: How do the breaks in the domain of h(x) relate to the zeros of f (x)?

Answer 1

Answer: Part A: The domain of f(x) is all real numbers, while the domain of g(x) is the interval (-, -1] [3, ]. The range of both f(x) and g(x) is all real values.

Part B: The breaks in the domain of h(x) correspond to the zeros of the polynomial function f(x), establishing a link between the two functions at those locations.

Step-by-step explanation: Part A: Comparing Domain and Range of f(x) and g(x)

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. The range of a function refers to the set of all possible output values (y-values) that the function can produce.

For the polynomial function f(x) = x^4 – 4x^3 – 2x^2 + 12x + 9, there are no restrictions on the type of input values (x) that can be used. Therefore, the domain of f(x) is all real numbers.

For the radical function g(x) = √(x^2 - 2x - 3), the expression inside the square root must be non-negative to ensure a real output. Therefore, the quadratic expression x^2 - 2x - 3 must be greater than or equal to zero. By solving this quadratic inequality, we find that the solutions are x ≤ -1 and x ≥ 3. This means that the domain of g(x) is the interval (-∞, -1] ∪ [3, ∞).

In terms of the range, both f(x) and g(x) are continuous functions, and their range extends to all real numbers. Therefore, the range of both f(x) and g(x) is also all real numbers.

Part B: Relationship between Breaks in Domain of h(x) and Zeros of f(x)

The function h(x) is a rational function, which means that its denominator cannot be equal to zero to avoid division by zero. The denominator of h(x) is x^2 - 2x - 3. The breaks in the domain of h(x) occur where the denominator is equal to zero.

To find the zeros of the denominator x^2 - 2x - 3, we solve the quadratic equation x^2 - 2x - 3 = 0. By factoring or using the quadratic formula, we find the solutions as x = -1 and x = 3. These are the same points where the domain of h(x) breaks.

In relation to the zeros of f(x), if we factor the polynomial x^4 – 4x^3 – 2x^2 + 12x + 9, we can find its roots (zeros). The zeros of f(x) are likely to be related to the points where the domain of h(x) breaks, which are x = -1 and x = 3.

In summary, the breaks in the domain of h(x) correspond to the zeros of f(x), suggesting a relationship between the two functions at these specific points.