What is the range of the function f(x) = (2x - 1) / (x + 3), where x is a real number and x ≠ -3?
Show the answer and explanation
Correct answer
A. All real numbers except 2
Principle or equation
For a rational function of the form (ax + b)/(cx + d), the range is all real numbers except a/c, provided c ≠ 0.
Why this answer is correct
Let y = (2x - 1)/(x + 3). Solve for x: y(x + 3) = 2x - 1 => yx + 3y = 2x - 1 => x(y - 2) = -1 - 3y => x = (-1 - 3y)/(y - 2). This is defined for all y except y = 2. Thus range is all real numbers except 2.
Example
For f(x) = (3x + 1)/(x - 2), the range is all real numbers except 3.
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