Answer:
[tex]g(x) = |\frac{1}{2}x - \frac{3}{2} | + 3[/tex]
Step-by-step explanation:
Given
[tex]y = |\frac{1}{2}x - 2| + 3[/tex]
Required
Translate the above one unit to the left
Replace y with f(x)
[tex]y = |\frac{1}{2}x - 2| + 3[/tex]
[tex]f(x) = |\frac{1}{2}x - 2| + 3[/tex]
When an absolute function is translated to the left, the resulting function is
[tex]g(x) = f(x - h)[/tex]
Because it is been translated 1 unit to the left, h = -1
[tex]g(x) = f(x - (-1))[/tex]
[tex]g(x) = f(x + 1)[/tex]
Calculating [tex]f(x+1)[/tex]
[tex]f(x+1) = |\frac{1}{2}(x+1) - 2| + 3[/tex]
Open bracket
[tex]f(x+1) = |\frac{1}{2}x + \frac{1}{2} - 2| + 3[/tex]
[tex]f(x+1) = |\frac{1}{2}x + \frac{1-4}{2} | + 3[/tex]
[tex]f(x+1) = |\frac{1}{2}x + \frac{-3}{2} | + 3[/tex]
[tex]f(x+1) = |\frac{1}{2}x - \frac{3}{2} | + 3[/tex]
Recall that
[tex]g(x) = f(x + 1)[/tex]
Hence;
[tex]g(x) = |\frac{1}{2}x - \frac{3}{2} | + 3[/tex]
