Problem

Given the rational function \( f(x) = \frac{(x-1)}{(x+2)} \), find the range of \( f(x) \).

Answer

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Answer

Therefore, the range of \( f(x) \) is all real numbers except \( f(-2) \).

Steps

Step 1 ：The range of a rational function \( f(x) = \frac{p(x)}{q(x)} \) is the set of all real numbers except the value \( a \) such that \( q(a) = 0 \).

Step 2 ：In the given function \( f(x) = \frac{(x-1)}{(x+2)} \), the denominator is \( x+2 \), so \( x+2 \neq 0 \), which implies that \( x \neq -2 \).

Step 3 ：Substituting \( x = -2 \) in \( f(x) \), we get \( f(-2) = \frac{(-2-1)}{(-2+2)} = \frac{-3}{0} \), which is undefined.

Step 4 ：Therefore, the range of \( f(x) \) is all real numbers except \( f(-2) \).

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