Question

6) Ethan uses the function f(x) = 2x2 + 2x + 4 to track the temperature of a substance over 8 hours. You know that f(3) = 28 meaning that the temperature of the substance is 28°C when you plug in a 3 for x. What is the domain of Ethan’s function?

Answer 1

The domain of Ethan's function f(x) = 2x² + 2x + 4 is all real numbers as it is a quadratic function.

Step-by-step explanation:

The domain of a function is the set of all possible input values (x-values) for which the function produces a valid output (y-value). In this case, the function f(x) = 2x² + 2x + 4 represents the temperature of a substance over time. Since temperature is a physical quantity and cannot be negative, the domain of the function must be restricted to non-negative values of x.

To determine the exact domain of the function, we can analyze the expression 2x² + 2x + 4. This expression is always non-negative for any real number x. Therefore, the domain of f(x) = 2x² + 2x + 4 is all real numbers x ≥ 0.

In mathematical notation, the domain of the function can be expressed as:

x ∈ ℝ

This indicates that the function is defined for all real numbers (ℝ) as long as the value of x is greater than or equal to 0 (x ≥ 0).

Answer 2

You have the quadratic function
`f(x)=2 x^(2) +2x+4`, and remember that all quadratic functions are parabolas opening at point called the vertex. Since you can plug any real number in our parabola here, its domain is all the real numbers, and this is true for all the quadratic functions of the form
`a x^(2) +bx+c`.
In interval notation the domain of the function
`f(x)=2 x^(2) +2x+4` is
(-∞, ∞)