The true about the domain and the range of the function is:
The domain is all real numbers, and the range is all real numbers
greater than or equal to -4 ⇒ 1st answer
Step-by-step explanation:
f(x) = (x + 2)(x + 6) is a quadratic function with 2 factors (x + 2) and (x + 6)
By multiplying its two factors we will find the form of the quadratic function
∵ (x)(x) = x²
∵ (x)(6) = 6x
∵ (2)(x) = 2x
∵ (2)(6) = 12
∴ f(x) = x² + 6x + 2x + 12
- By adding like terms
∴ f(x) = x² + 8x + 12
The quadratic function represented graphically by a parabola
Look to the attached figure
The x-coordinate of the vertex point of the parabola h = [tex]\frac{-b}{a}[/tex]
where b is the coefficient of x and a is the coefficient of x²
∵ f(x) = x² + 8x + 12
∴ a = 1 and b = 8
∴ h = [tex]\frac{-8}{2*1}=-4[/tex]
The y-coordinate of the vertex point is k = f(h)
∵ h = -4
∴ k = f(-4)
∴ k = (-4)² + 8(-4) = 12 = 16 - 32 + 12
∴ k = -4
∴ The vertex point of the parabola is (-4 , -4)
∵ The parabola is opened upward
∴ Its vertex is minimum point
∴ The minimum value of f(x) is y = -4
∵ The domain of the function is the values of x
∵ The range of the function is the values of y corresponding to the
values of x
∵ x can be any real numbers
∴ x ∈ R, where R is the set of real numbers
∴ The domain of f(x) is all real numbers
∵ The minimum value of f(x) is y = -4
∴ y can be any real number greater than or equal to -4
∴ y ≥ -4
∴ The range is all real number greater than or equal to -4
The true about the domain and the range of the function is:
The domain is all real numbers, and the range is all real numbers
greater than or equal to -4
Learn more:
You can learn more about quadratic function in brainly.com/question/1332667
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Using the vertex of a quadratic equation, it is found that the correct option is:
The domain is all real numbers, and the range is all real numbers greater than or equal to -4.
A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The vertex is given by:
[tex](x_v, y_v)[/tex]
In which:
[tex]x_v = -\frac{b}{2a}[/tex]
[tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]
The domain is all real numbers. As for the range, considering the coefficient a, we have that:
In this problem, the equation is given by:
f(x) = (x + 2)(x + 6)
In standard form:
f(x) = x^2 + 8x + 12.
Hence the coefficients are a = 1 > 0, b = 8, c = 12 and:
[tex]y_v = -\frac{8^2 - 4(1)(12)}{4} = -4[/tex]
Hence the correct option is:
The domain is all real numbers, and the range is all real numbers greater than or equal to -4.
More can be learned about the vertex of a quadratic equation at the vertex of at https://brainly.com/question/24737967
