To find [tex]\( f(x-1) \)[/tex], let's start by analyzing the given function [tex]\( f(x+1) = x + 2 \)[/tex].
We need to express [tex]\( f(x-1) \)[/tex] in terms of the same function. First, let’s make a substitution to simplify the problem.
Let’s set:
[tex]\[ y = x - 1 \][/tex]
So if [tex]\( y = x - 1 \)[/tex], then we can solve for [tex]\( x \)[/tex]:
[tex]\[ x = y + 1 \][/tex]
Now, substitute [tex]\( x = y + 1 \)[/tex] back into the function [tex]\( f(x+1) \)[/tex]:
[tex]\[ f((y + 1) + 1) = (y + 1) + 2 \][/tex]
This simplifies the original equation to:
[tex]\[ f(y + 2) = y + 3 \][/tex]
Next, we need to translate this back to the function we are finding, which is [tex]\( f(y) \)[/tex]. According to our substitution:
[tex]\[ y = x - 1 \][/tex]
So if we now consider [tex]\( y + 2 = (x - 1) + 2 = x + 1 \)[/tex], thus:
[tex]\[ f(x + 1) = x + 2 \][/tex]
From [tex]\( f(y + 2) = y + 3 \)[/tex], and since [tex]\( y = x - 1 \)[/tex], this translates to:
[tex]\[ f((x - 1) + 1) = x \][/tex]
So, we find that:
[tex]\[ f(x - 1) = x \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{x} \][/tex]
