Is the inverse of a function #(x + 2)^2 - 4# a function?

Question

Elijah Allen · Accepted Answer

It depends on the domain you pick. If #x# varies over the entire real line, the answer is "no". If #x# varies over a restricted domain, the answer can be "yes".

If you let #f# be the function, defined for all #x\in RR# by the formula #f(x)=(x+2)^2-4#, then this function does not pass the horizontal line test (lots of horizontal lines go through its graph, which is a parabola, more than once) and does not have an inverse function. You could say, however, that it has an "inverse relation ". If you reflect the (full) graph of #y=f(x)=(x+2)^2-4# across the diagonal line #y=x#, you'll get a sideways parabola that does not pass the vertical line test (lots of vertical lines go through its graph more than once). Such a graph cannot be the graph of a function, which must have a unique output for each input. However, mathematicians still give a name to such an object. They call it a relation. On the other hand, if you restrict the domain of the original function #f(x)=(x+2)^2-4# to, for instance, #x\geq -2#, then the graph is just the right half of the original parabola and passes the horizontal line test, and therefore has an inverse function. If you solve the equation #y=(x+2)^2-4# for #x \geq -2#, you'll get a formula for the inverse function with #y# as the independent variable: #x=f^{-1}(y)=-2+sqrt{y+4}#. If you decide to switch the variables around to write #y=f^{-1}(x)=-2+sqrt{x+4}# and graph this, you'll see the reflection property. On the other hand once again, if you restrict the domain of the original function #f(x)=(x+2)^2-4# to, for instance, #x\leq -2#, then the graph is just the left half of the original parabola and passes the horizontal line test, and therefore has an inverse function. If you solve the equation #y=(x+2)^2-4# for #x \leq -2#, you'll get a formula for the inverse function with #y# as the independent variable: #x=f^{-1}(y)=-2-sqrt{y+4}#. If you decide to switch the variables around to write #y=f^{-1}(x)=-2-sqrt{x+4}# and graph this, you'll see the reflection property again (but for the other half of the original parabola).

Serenity Johnson · Answer

undefined Yes, the inverse of a function is a function if and only if the original function is one-to-one, meaning each element in its domain maps to a unique element in its range. To determine if the inverse of the function $ (x + 2)^2 - 4 $ is a function, we need to check if the original function is one-to-one.

The given function is $ f(x) = (x + 2)^2 - 4 $. To determine if it's one-to-one, we can check if it passes the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one, and its inverse won't be a function.

However, the given function $ f(x) = (x + 2)^2 - 4 $ is a quadratic function. Quadratic functions are not one-to-one because they have a vertex which is the lowest or highest point of the parabola and it will have a horizontal line of symmetry. Therefore, the function $ (x + 2)^2 - 4 $ is not one-to-one, and its inverse will not be a function.