Question

What are the domain and range of the function f of x is equal to the quantity x squared plus 6x plus 8 end quantity divided by the quantity x plus 4 end quantity? D: x ∊ ℝ ; R: y ≠ 0 D: x ≠ −4; R: y ∊ ℝ D: x ∊ ℝ ; R: y ≠ 2 D: x ∊ ℝ ; R: y ∊ ℝ

Answer 1

The domain of the function f(x) = (x² + 6x + 8) / (x + 4) is x ≠ -4, and the range is y ≠ 2.

Step-by-step explanation:

The domain and range of a function can be found by looking at the limitations set by the function's definition itself. For the function f(x) = (x² + 6x + 8) / (x + 4), the domain is restricted due to a potential division by zero when x = -4. Therefore, we can express the domain as D: x ∈ ℝ .

Next, we can determine the range. By simplifying the function, it becomes apparent that the function can never attain a certain value. Factoring the numerator gives us (x + 4)(x + 2) / (x + 4), which simplifies to x + 2 for all x except -4. The value x cannot be is 2, because this would imply the factor (x + 4) was equal to zero, which is not allowed. Consequently, the range is R: y ∈ ℝ .

Therefore, the correct representation of the domain is x ≠ -4; and the range is y ∈ ℝ .

Answer 2

The domain of the function f(x) = (x^2 + 6x + 8) / (x + 4) is all real numbers except x = -4, and the range is all real numbers, making D: x ≠ −4 and R: {y ∊ ℝ}.

Step-by-step explanation:

We are given a function f(x) which is equal to the quantity x squared plus 6x plus 8 divided by the quantity x plus 4.

To find the domain and range, we first need to factor the numerator if possible and simplify the expression.

The numerator x2+6x+8 can be factored into (x+4)(x+2).

The denominator is x+4, so the function simplifies to f(x) = x + 2 when x ≠ -4 because we cannot divide by zero (division by zero is undefined).

Therefore, the domain of f(x) is all real numbers except for x = -4.

This is because the value of x cannot be -4 since it would make the denominator of the original function zero.

Hence, the domain is D: x ∊ ℝ .

The simplified function f(x) = x + 2 is a straight line which means its range is all real numbers.

There are no restrictions on the value of f(x); hence, the range is all real numbers, so R: {y ∊ ℝ}.