Problem

Given the rational function f(x)=(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)=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)=(x−1)(x+2), the denominator is x+2, so x+2≠0, which implies that x≠−2.

Step 3 ：Substituting x=−2 in f(x), we get f(−2)=(−2−1)(−2+2)=−30, which is undefined.

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

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