Show that the function f(x) = x-2x+1, x ≠ – 1 is increasing - Mathematics and Statistics

Question

Show that the function f(x) = $\frac{x - 2}{x + 1}$ , x ≠ – 1 is increasing

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Solution

f(x) = $\frac{x - 2}{x + 1}$ for x ≠ – 1

For function to be increasing, f'(x) > 0

Then, f'(x) = $\frac{(x + 1) \frac{d}{d x} (x - 2) - (x - 2) \frac{d}{d x} (x + 1)}{{(x + 1)}^{2}}$

= $\frac{(x + 1) - (x - 2)}{{(x + 1)}^{2}}$

= $\frac{x + 1 - x + 2}{{(x + 1)}^{2}}$

= $\frac{3}{{(x + 1)}^{2}} > 0$ .......[∵ (x + 1) ≠ 0, (x + 1)² > 0]

Thus, f(x) is an increasing function for x ≠ – 1.

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