Question

Determine the values of x for which f(x) = `(x - 3)/(x + 1)`, x ≠ −1 is an increasing function.

Answer 1

Step 1: Find the derivative f′(x)

Use the quotient rule:

`[u/v]^′ = (u^′v - uv^′)/v^2`, where u = x − 3 and v = x + 1

f′(x) = `((1)(x + 1) - (x - 3)(1))/(x + 1)^2`

= `(x + 1 - x + 3)/(x + 1)^2`

= `4/(x + 1)^2`

Step 2: Set the condition for an increasing function

A function is strictly increasing where its derivative is positive (f)′(x) > 0:

`4/(x + 1)^2` > 0

Since the numerator (4) and the denominator (x + 1)² are always positive for all x in the domain (x ≠ −1)

Therefore, f′(x) is positive for all values in the domain of the function.

The function is increasing for all x ∈ R, x ≠ −1, or in interval notation, (−∞, −1) ∪ (−1, ∞).