To determine which function results in a horizontal compression of [tex]\( y = \frac{1}{x} \)[/tex] by a factor of 6, let us review what horizontal compression means.
1. Understanding Horizontal Compression:
- To horizontally compress a function by a factor of [tex]\( k \)[/tex], you replace [tex]\( x \)[/tex] in the original function [tex]\( f(x) \)[/tex] with [tex]\( \frac{x}{k} \)[/tex].
- For example, if [tex]\( f(x) = \frac{1}{x} \)[/tex] and you want to horizontally compress it by a factor of 6, you would replace [tex]\( x \)[/tex] with [tex]\( \frac{x}{6} \)[/tex].
2. Applying Horizontal Compression:
- Start with the original function: [tex]\( y = \frac{1}{x} \)[/tex].
- Compress horizontally by a factor of 6 by replacing [tex]\( x \)[/tex] with [tex]\( \frac{x}{6} \)[/tex]:
[tex]\[
y = \frac{1}{\frac{x}{6}}
\][/tex]
3. Simplifying the Function:
- Simplify [tex]\( \frac{1}{\frac{x}{6}} \)[/tex]:
[tex]\[
y = \frac{1}{\frac{x}{6}} = \frac{1 \cdot 6}{x} = \frac{6}{x}
\][/tex]
Thus, the function [tex]\( y = \frac{6}{x} \)[/tex] is the result of a horizontal compression of [tex]\( y = \frac{1}{x} \)[/tex] by a factor of 6.
Comparing this to the options:
- [tex]\( y = \frac{1}{6x} \)[/tex] is incorrect as it represents a horizontal stretch.
- [tex]\( y = -\frac{1}{6x} \)[/tex] is incorrect as it also represents a stretch and a reflection.
- [tex]\( y = \frac{6}{x} \)[/tex] is correct.
- [tex]\( y = -\frac{6}{x} \)[/tex] is incorrect as it includes an additional reflection.
Therefore, the correct option is:
[tex]\[ \boxed{y = \frac{6}{x}} \][/tex]
This corresponds to option 3.
