How do you find the inverse of #f(x)= -|x+1|+4#?

Question

Eli Assouline · Accepted Answer

The function #f(x)# does not have an inverse.

Given a function of x such that #y = f(x)#, the function has an inverse, if and only if, there is one and only one value of x that produces a given value of y. Except for the point #(-1,4)# there are two values of x that produce a given value of y, therefore, there is no inverse.

Avery Alexander · Answer

undefined To find the inverse of $ f(x) = -|x+1| + 4 $:

1. Replace $ f(x) $ with $ y $: $ y = -|x+1| + 4 $.

2. Swap $ x $ and $ y $: $ x = -|y+1| + 4 $.

3. Solve for $ y $:
   - Subtract 4 from both sides: $ x - 4 = -|y+1| $.
   - Multiply both sides by -1: $ 4 - x = |y+1| $.

4. Solve for $ |y+1| $:
   - If $ 4 - x \geq 0 $, then $ 4 - x = y + 1 $.
     - Subtract 1 from both sides: $ 3 - x = y $.
   - If $ 4 - x < 0 $, then $ 4 - x = -(y + 1) $.
     - Multiply both sides by -1: $ x - 4 = y + 1 $.
     - Subtract 1 from both sides: $ x - 5 = y $.

5. Combine both cases:
   - $ y = \begin{cases} 3 - x & \text{if } x \leq 4 \ x - 5 & \text{if } x > 4 \end{cases} $.

Therefore, the inverse function is $ f^{-1}(x) = \begin{cases} 3 - x & \text{if } x \leq 4 \ x - 5 & \text{if } x > 4 \end{cases} $.