Problem 30 - Calculus of a Single Variable

Question

Show that \(f\) is strictly monotonic on the given interval and therefore has an inverse function on that interval. \(f(x)=|x+2|, \quad[-2, \infty)\)

Step-by-Step Solution

Answer

The function \(f(x) = |x+2|\) is strictly increasing on the interval [-2, \(\infty\)], and therefore it is strictly monotonic on that interval. Hence, it has an inverse function on the interval [-2, \(\infty\)].

Step 1: Understand function behaviour

Observe that the function \(f(x) = |x+2|\) is the absolute function shifted to the left by 2 units. Recall that the absolute function \(y = |x|\) is a V-shaped curve which is symmetrical about the y-axis and is always non-negative. It is decreasing for \(x < 0\) and increasing for \(x > 0\). Therefore, the function \(f(x) = |x+2|\) is decreasing for \(x < -2\) and increasing for \(x > -2\).

Step 2: Determine function behaviour on the given interval

The given interval is [-2, \(\infty\)]. For \(x > -2\), \(f(x) = |x+2|\) is an increasing function; for \(x = -2\), \(f(-2) = 0\). Therefore, on the interval [-2, \(\infty\)], \(f(x)\) is strictly increasing.

Step 3: Conclude

Since \(f(x) = |x+2|\) is strictly increasing on the interval [-2, \(\infty\)], it is strictly monotonic on that interval. Hence it has an inverse function on that interval.