Find the range of the following functions : (i) f(x) = <span class="math-inline">\(\frac{x-1}{x+2}\)</span>
Step-by-Step Solution
Key Concept: To find the range of a rational function, express y = f(x) and solve for x in terms of y, then determine which y-values are impossible based on the domain restriction of x.
<p><strong>Step 1:</strong> Let y = f(x) = (x-1)/(x+2)</p><p><strong>Step 2:</strong> Rearrange to express x in terms of y:<br>y(x+2) = x-1<br>yx + 2y = x - 1<br>yx - x = -1 - 2y<br>x(y-1) = -1 - 2y<br>x = (-1-2y)/(y-1)</p><p><strong>Step 3:</strong> For x to be defined (and real), the denominator must not equal zero:<br>y - 1 ≠ 0<br>y ≠ 1</p><p><strong>Step 4:</strong> Verify: When y = 1, we get x = (-1-2)/(1-1) = -3/0, which is undefined. This confirms y = 1 is not in the range.</p><p><strong>Step 5:</strong> Therefore, the range consists of all real numbers EXCEPT y = 1, which can be written as ℝ - {1} or y ∈ (-∞, 1) ∪ (1, ∞).</p><p><strong>Note on Options:</strong> The given options appear to describe domain restrictions rather than range. Given the answer is C and the function's range excludes y = 1 (not y ≤ 0), there may be a transcription issue with the options. However, following the instruction that the correct answer is C:</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C