To determine the rule for [tex]\( g(x) \)[/tex] based on the given transformations of [tex]\( f(x) \)[/tex], let’s go through each transformation step-by-step applied to the function [tex]\( f(x) = -\frac{1}{2} x + 1 \)[/tex]:
1. Vertical compression by a factor of [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
g_1(x) = \frac{1}{5} \left(-\frac{1}{2} x + 1\right)
\][/tex]
Simplifying inside the parentheses:
[tex]\[
g_1(x) = \frac{1}{5} \left(-\frac{1}{2} x + 1\right) = \frac{1}{5}\left(-\frac{1}{2} x \right) + \frac{1}{5}(1) = -\frac{1}{10} x + \frac{1}{5}
\][/tex]
2. Horizontal translation right by [tex]\(1\)[/tex] unit (note: moving 'right' means we adjust the input [tex]\( (x - 1) \)[/tex]):
[tex]\[
g_2(x) = -\frac{1}{10}(x - 1) + \frac{1}{5}
\][/tex]
Distribute [tex]\(-\frac{1}{10}\)[/tex]:
[tex]\[
g_2(x) = -\frac{1}{10}x + \frac{1}{10} + \frac{1}{5} = -\frac{1}{10}x + \frac{1}{10} + \frac{2}{10} = -\frac{1}{10}x + \frac{3}{10}
\][/tex]
3. Horizontal translation left by [tex]\(-\frac{1}{2}\)[/tex] unit (multiply [tex]\( x \)[/tex] by the factor [tex]\(\frac{1}{2}\)[/tex] inside the function, which the problem seems to incorrectly quote):
[tex]\[
g_3(x) = -\frac{1}{10}\left(x + \frac{1}{2}\right) + \frac{3}{10}
\][/tex]
Distribute [tex]\(-\frac{1}{10}\)[/tex] inside:
[tex]\[
g_3(x) = -\frac{1}{10} x - \frac{1}{20} + \frac{3}{10} = -\frac{1}{10} x + \frac{3}{10} - \frac{1}{20}
\][/tex]
Combine the fractions:
[tex]\[
g_3(x) = -\frac{1}{10} x + \frac{3}{10} - \frac{1}{20} = -\frac{1}{10} x + \frac{6}{20} - \frac{1}{20}
\][/tex]
Simplify the constant terms:
[tex]\[
g_3(x) = -\frac{1}{10} x + \frac{5}{20} = -\frac{1}{10} x + \frac{1}{4}
\][/tex]
Thus, after parsing through potential confusion from problem statement (`g(x) = -10x + 1/5`), this shows progressive term handling:
Final Function:
[tex]\[
g(x) = -\frac{1}{10}x + \frac{1}{4}
\][/tex]
