How do you find the inverse of #f(x) = | x | - 3# and is it a function?

Question

How do you find the inverse of  #f(x) = | x | - 3# and is it a function?

Elias Ackerman · Accepted Answer

The inverse function of #f# does not exist.

Assuming that, # f : RR to RR : x to f(x)=|x|-3#, we find, # f(-1)=|-1|-3=1-3=-2, and, &, # # f(1)=|1|-3=-2#. # rArr f(-1)=f(1)," but, "-1 != 1#. Thus, #f# is not an injection. We know that, #"The inverse function of a given function exists"# #iff" f is bijective, i.e., injective and surjectvie, both"#. Hence, the inverse function of #f# does not exist.

Theodore Amidon · Answer

undefined To find the inverse of the function $ f(x) = |x| - 3 $, follow these steps:

1. Replace $ f(x) $ with $ y $.
2. Swap the roles of $ x $ and $ y $.
3. Solve for $ y $.
4. Replace $ y $ with $ f^{-1}(x) $.

Doing this, we get:

$$ x = |y| - 3 $$

Now, solve for $ y $:

$$ x + 3 = |y| $$

Since the absolute value function can be rewritten as two separate equations, one for when $ y $ is positive and one for when $ y $ is negative, we have:

When $ y $ is positive:

$$ y = x + 3 $$

When $ y $ is negative:

$$ y = -x - 3 $$

Therefore, the inverse function, $ f^{-1}(x) $, consists of two separate equations:

$$ f^{-1}(x) = \begin{cases} 
      x + 3 & \text{if } x \geq 0 \
      -x - 3 & \text{if } x < 0 
   \end{cases} $$

As for whether it is a function, yes, the inverse of $ f(x) = |x| - 3 $ is a function. Both equations in the inverse function pass the vertical line test, meaning each input value (or $ x $ value) corresponds to exactly one output value (or $ y $ value). Therefore, it is a function.