Answer :
To determine which function shows the reciprocal parent function, [tex]\(F(x) = \frac{1}{x}\)[/tex], vertically compressed, let's first understand what vertical compression means in this context.
A vertical compression of a function occurs when you multiply the function by a constant factor between 0 and 1 (exclusive). For the reciprocal function [tex]\(F(x) = \frac{1}{x}\)[/tex], if we multiply it by such a factor, the function looks like this:
[tex]\[ G(x) = \frac{1}{k \cdot x}, \text{ with } 0 < k < 1. \][/tex]
Let's examine the given options to find which one is vertically compressed:
Option A: [tex]\(G(x) = \frac{1}{27x}\)[/tex]
In this function, the constant factor multiplying [tex]\(x\)[/tex] is [tex]\(27\)[/tex]. When the function is written as [tex]\(\frac{1}{27 \cdot x}\)[/tex], we see that [tex]\(k = 27\)[/tex]. However, for vertical compression, [tex]\(k\)[/tex] should be between 0 and 1. Here, [tex]\(27\)[/tex] is greater than [tex]\(1\)[/tex], not satisfying vertical compression directly, but when considering the reciprocal case, it could still reflect vertical compression.
Option B: [tex]\(G(x) = \frac{6}{x}\)[/tex]
Here, the function [tex]\(\frac{6}{x}\)[/tex] is essentially [tex]\(6 \times \frac{1}{x}\)[/tex]. Multiplying the parent reciprocal function [tex]\(F(x) = \frac{1}{x}\)[/tex] by [tex]\(6\)[/tex] actually represents a vertical stretch, not a compression, because [tex]\(6 > 1\)[/tex].
Option C: [tex]\(G(x) = \frac{1}{x-9}\)[/tex]
In this function, the expression in the denominator is shifted by [tex]\(9\)[/tex], indicating a horizontal translation. This is neither a vertical stretch nor a compression, and it's a transformation of a different nature (horizontal shift).
Option D: [tex]\(G(x) = \frac{1}{x} + 3\)[/tex]
Here, [tex]\(\frac{1}{x} + 3\)[/tex] adds [tex]\(3\)[/tex] to the [tex]\(\frac{1}{x}\)[/tex] function, which represents a vertical shift up by [tex]\(3\)[/tex]. This also doesn't represent vertical compression or stretch but a vertical translation instead.
Given these considerations – the correct interpretation requires explicit recognition that vertical compression for the reciprocal function must appear like a multiplication of the entire denominator by a value indicating scaling factor [tex]\(k\)[/tex] less than one through direct reciprocal means as a case-specific interpretation:
Therefore, the function that represents the vertically compressed reciprocal parent function is: [tex]\(G(x) = \frac{1}{27x}\)[/tex]—so the correct answer is option [tex]\(A\)[/tex].
A vertical compression of a function occurs when you multiply the function by a constant factor between 0 and 1 (exclusive). For the reciprocal function [tex]\(F(x) = \frac{1}{x}\)[/tex], if we multiply it by such a factor, the function looks like this:
[tex]\[ G(x) = \frac{1}{k \cdot x}, \text{ with } 0 < k < 1. \][/tex]
Let's examine the given options to find which one is vertically compressed:
Option A: [tex]\(G(x) = \frac{1}{27x}\)[/tex]
In this function, the constant factor multiplying [tex]\(x\)[/tex] is [tex]\(27\)[/tex]. When the function is written as [tex]\(\frac{1}{27 \cdot x}\)[/tex], we see that [tex]\(k = 27\)[/tex]. However, for vertical compression, [tex]\(k\)[/tex] should be between 0 and 1. Here, [tex]\(27\)[/tex] is greater than [tex]\(1\)[/tex], not satisfying vertical compression directly, but when considering the reciprocal case, it could still reflect vertical compression.
Option B: [tex]\(G(x) = \frac{6}{x}\)[/tex]
Here, the function [tex]\(\frac{6}{x}\)[/tex] is essentially [tex]\(6 \times \frac{1}{x}\)[/tex]. Multiplying the parent reciprocal function [tex]\(F(x) = \frac{1}{x}\)[/tex] by [tex]\(6\)[/tex] actually represents a vertical stretch, not a compression, because [tex]\(6 > 1\)[/tex].
Option C: [tex]\(G(x) = \frac{1}{x-9}\)[/tex]
In this function, the expression in the denominator is shifted by [tex]\(9\)[/tex], indicating a horizontal translation. This is neither a vertical stretch nor a compression, and it's a transformation of a different nature (horizontal shift).
Option D: [tex]\(G(x) = \frac{1}{x} + 3\)[/tex]
Here, [tex]\(\frac{1}{x} + 3\)[/tex] adds [tex]\(3\)[/tex] to the [tex]\(\frac{1}{x}\)[/tex] function, which represents a vertical shift up by [tex]\(3\)[/tex]. This also doesn't represent vertical compression or stretch but a vertical translation instead.
Given these considerations – the correct interpretation requires explicit recognition that vertical compression for the reciprocal function must appear like a multiplication of the entire denominator by a value indicating scaling factor [tex]\(k\)[/tex] less than one through direct reciprocal means as a case-specific interpretation:
Therefore, the function that represents the vertically compressed reciprocal parent function is: [tex]\(G(x) = \frac{1}{27x}\)[/tex]—so the correct answer is option [tex]\(A\)[/tex].
