Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To determine the range of the function [tex]\( f(x) = (x - 4)(x - 2) \)[/tex], we need to analyze its properties as a quadratic function. Let's go through the steps:
1. Rewrite the function in standard form:
First, expand the given function:
[tex]\[ f(x) = (x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8 \][/tex]
Thus, the function is [tex]\( f(x) = x^2 - 6x + 8 \)[/tex].
2. Identify the shape and direction of the parabola:
The quadratic function [tex]\( f(x) = x^2 - 6x + 8 \)[/tex] is a parabola that opens upwards because the coefficient of [tex]\( x^2 \)[/tex] (a = 1) is positive.
3. Find the vertex of the parabola:
The vertex form of a parabolic function [tex]\( ax^2 + bx + c \)[/tex] gives the x-coordinate of the vertex as [tex]\( \frac{-b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex], so:
[tex]\[ x_{\text{vertex}} = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3 \][/tex]
4. Calculate the y-coordinate of the vertex:
Substitute [tex]\( x = 3 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3) = (3 - 4)(3 - 2) = (-1)(1) = -1 \][/tex]
Therefore, the vertex of the parabola is at [tex]\( (3, -1) \)[/tex].
5. Determine the range of the function:
Since the parabola opens upwards, the vertex represents the minimum value of the function. Consequently, the y-coordinate of the vertex is the minimum value of [tex]\( f(x) \)[/tex]. Thus, the range of [tex]\( f(x) \)[/tex] is all real numbers greater than or equal to -1.
Therefore, the correct answer is:
[tex]\[ \text{The range of the function is } \text{all real numbers greater than or equal to -1.} \][/tex]
1. Rewrite the function in standard form:
First, expand the given function:
[tex]\[ f(x) = (x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8 \][/tex]
Thus, the function is [tex]\( f(x) = x^2 - 6x + 8 \)[/tex].
2. Identify the shape and direction of the parabola:
The quadratic function [tex]\( f(x) = x^2 - 6x + 8 \)[/tex] is a parabola that opens upwards because the coefficient of [tex]\( x^2 \)[/tex] (a = 1) is positive.
3. Find the vertex of the parabola:
The vertex form of a parabolic function [tex]\( ax^2 + bx + c \)[/tex] gives the x-coordinate of the vertex as [tex]\( \frac{-b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex], so:
[tex]\[ x_{\text{vertex}} = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3 \][/tex]
4. Calculate the y-coordinate of the vertex:
Substitute [tex]\( x = 3 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3) = (3 - 4)(3 - 2) = (-1)(1) = -1 \][/tex]
Therefore, the vertex of the parabola is at [tex]\( (3, -1) \)[/tex].
5. Determine the range of the function:
Since the parabola opens upwards, the vertex represents the minimum value of the function. Consequently, the y-coordinate of the vertex is the minimum value of [tex]\( f(x) \)[/tex]. Thus, the range of [tex]\( f(x) \)[/tex] is all real numbers greater than or equal to -1.
Therefore, the correct answer is:
[tex]\[ \text{The range of the function is } \text{all real numbers greater than or equal to -1.} \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.