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Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing

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#### Solution

f(x) = `(x - 2)/(x + 1)` for x ≠ – 1

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

Then, f'(x) = `((x + 1)"d"/("d"x)(x - 2) - (x - 2)"d"/("d"x)(x + 1))/(x + 1)^2`

= `((x + 1) - (x - 2))/(x + 1)^2`

= `(x + 1 - x + 2)/(x + 1)^2`

= `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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