To determine the equation of the new function after horizontally stretching the square root parent function [tex]\( F(x) = \sqrt{x} \)[/tex] by [tex]\( \frac{3}{4} \)[/tex] units, we need to understand how horizontal stretching affects the function.
For a horizontal stretch by a factor of [tex]\( \frac{b}{a} \)[/tex]:
1. The transformation formula is [tex]\( G(x) = F\left(\frac{a}{b} x \right) \)[/tex].
Given a horizontal stretch by [tex]\( \frac{3}{4} \)[/tex], we interpret this as stretching by a reciprocal factor of [tex]\( \frac{4}{3} \)[/tex]. Thus:
2. [tex]\( a = 4 \)[/tex] and [tex]\( b = 3 \)[/tex], yielding the horizontal stretch factor [tex]\( \frac{4}{3} \)[/tex].
Applying this to the parent function [tex]\( F(x) = \sqrt{x} \)[/tex]:
3. We transform it as [tex]\( G(x) = F\left(\frac{4}{3} x \right) \)[/tex].
Substitute the parent function [tex]\( \sqrt{x} \)[/tex] into this transformation:
4. [tex]\( G(x) = \sqrt{\frac{4}{3}x} \)[/tex].
Upon simplification, the equation for the new function after the stretch is:
5. [tex]\( G(x) = \sqrt{\frac{4}{3}x} \)[/tex].
Thus, the correct choice from the given options is:
D. [tex]\( G(x) = \sqrt{\frac{3}{4} x} \)[/tex].
After verifying our work, the correct interpretation is indeed consistent with the horizontal stretch by [tex]\( \frac{3}{4} \)[/tex] resulting in [tex]\( G(x) = \sqrt{\frac{4}{3}x} \)[/tex]. Hence, the accurate transformation and resulting equation are:
D. [tex]\( G(x) = \sqrt{\frac{4}{3} x} \)[/tex].
