Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval.
f(x) = abs. val. (x + 2), [-2, infinity)
Please do not change screen-names. It is easier for Erica/Brandon because reference can be made to a previous problem.
http://www.jiskha.com/display.cgi?id=1291619172
For this problem, you do not need to find f'(x), because it consists of two straight lines that intersect. A single straight line is monotonic if it is not horizontal.
Treat f(x) as two separate parts:
f(x)=-(x+2) if x<-2, and
f(x)=x+2 if x≥-2.
So if the domain is given in [2,∞], it consists of a straight line and therefore has an inverse.
Follow the steps set out in your previous question to find the inverse, if necessary, or as practice.
http://www.jiskha.com/display.cgi?id=1291623706
To show that a function f is strictly monotonic on a given interval, we need to demonstrate that the function is either strictly increasing or strictly decreasing on that interval.
For f(x) = |x + 2| on the interval [-2, infinity), we can prove its monotonicity as follows:
1. Show that it is strictly increasing:
To prove f(x) is strictly increasing, we need to show that for any two values a and b on the interval [-2, infinity) such that a < b, f(a) < f(b).
Let's consider two cases:
- Case 1: a, b >= -2 and a < b:
For this case, f(a) = |a + 2| and f(b) = |b + 2|.
Since both a and b are greater than or equal to -2, the absolute values will be non-negative. Therefore, we can rewrite the inequality as a + 2 < b + 2.
By canceling out 2 from both sides, we have: a < b.
This is true by the assumption of the case.
The condition f(a) < f(b) holds, verifying that f(x) is strictly increasing on the interval.
- Case 2: a = -2 and b > -2:
Here, f(a) = |a + 2| = 0 and f(b) = |b + 2|.
Since a = -2, we have f(a) = 0, and since b > -2, we have f(b) > 0.
Since 0 < f(b), the condition f(a) < f(b) holds, verifying that f(x) is strictly increasing on the interval.
Thus, we have shown that f(x) = |x + 2| is strictly increasing on the interval [-2, infinity).
2. Conclude the existence of an inverse function:
Since we have proven that f(x) is strictly monotonic (strictly increasing in this case) on the interval [-2, infinity), we can conclude that f(x) has an inverse function on that interval. The existence of an inverse function for strictly monotonic functions is a result of the fundamental property of strictly monotonic functions.
Therefore, the function f(x) = |x + 2| has an inverse function on the interval [-2, infinity).