What is the reverse of 1

Power functions: setting up the inverse function easily explained

definition

Learn what an inverse function is and how you can calculate an inverse function.

Inverse function

Inverse functions, as the name suggests, assign the variables in reverse. This means that the $ x $ value is swapped for the $ y $ value. This is only possible if there is only one $ x $ value for each function value $ (y) $. You can graphically create the inverse function by mirroring the function at the bisector, i.e. at the function $ g (x) = x $.

The inverse function of the function $ f (x) $ is marked with $ f ^ {\ textcolor {red} {- 1}} (x) $. The superscript $ \ textcolor {red} {- 1} $ is the symbol for the inverse function.
To create an inverse function, the function must first be converted to $ x $. Then $ x $ and $ y $ are swapped, and the definition and value sets are swapped.

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Procedure: Form the inverse function

  • Solve the function for $ x $.
  • Swap $ x $ and $ y $.
  • Let's look at two examples:

    Click here to expand$ y = 3x ^ 2 + 5 $

    Here we have to restrict the domain because the image is a square parabola that is not unique.

    The parabola has its vertex on the $ y $ axis. It is thus reversible for $ x ≥ 0 $, for example. This parabola is clear. The domain of definition for this function is therefore all real numbers that are greater than or equal to zero. The range of values ​​is made up of all real $ y $ values ​​that are greater than or equal to 5, because the parabola is open at the top and its vertex is 5 on the $ y $ axis.

    Definition area: D$ f $: $ x $ ∈ & Ropf ;, $ x $ ≥0

    Range of values: W$ f $: $ y $ ∈ & Ropf ;, $ y $ ≥5

    1. Solve the function for $ x $.

    $ y = 3x ^ 2 + 5 ~~~~~~~~~~~~~~~~~~~~~~~ | -5 $
    $ y-5 = 3x ^ 2 ~~~~~~~~~~~~~~~~~~~~~~~~ |: 3 $
    $ \ frac {y-5} {3} = x ^ 2 ~~~~~~~~~~~~~~~~~~ | \ sqrt {~~} $
    $ \ sqrt {\ frac {y-5} {3}} = x $

    2. Swap $ x $ and $ y $.

    $ \ sqrt {\ frac {x-5} {3}} = y $ or $ y = \ sqrt {\ frac {x-5} {3}} $

    $ f ^ {- 1} (x) = \ sqrt {\ frac {x-5} {3}} $

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    Here we form the inverse function for $ x $ ≥ 0.

    The example is available for the entire domain on How do you form an inverse function?

    $ f (x) = 5x ^ 3 $

    1. Solve the function for $ x $.

    $ y = 5x ^ 3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ |: 5 $

    $ \ frac {y ~} {5 ~} = x ^ 3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ | \ sqrt [3] {~ ~} $

    $ \ sqrt [3] {\ frac {y ~} {5 ~}} = x $

    2. Swap $ x $ and $ y $.

    $ f ^ {- 1} (x) = \ sqrt [3 ~] {\ frac {x ~} {5 ~}} $

    Power function

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    For every integer n, $ f (x) = x ^ \ textcolor {red} {n} $ is a power function.

    Power functions with positive exponents always run through the origin. In this text we only look at the inverse functions of such power functions.

    Figure: Graphs of power functions with natural exponents

    What do the inverse functions of such power functions with positive exponents look like?

    Inverse functions of power functions

    The inverse function of the power function $ f (x) = x ^ 3 $ is to be formed. We proceed as described above:

    Click here to expand

    Here, too, we form the inverse function for x≥0. We restrict the range of definition here, since root functions for negative values ​​are not explained.

    1. Solve the function for $ x $:

    $ y = x ^ 3 ~~~~~~~ | \ sqrt [3] {~~} $
    $ \ sqrt [3] {y} = x $

    2. Swap $ x $ and $ y $:

    $ y = \ sqrt [3] {x} $ or $ f ^ {- 1} (x) = y = \ sqrt [3] {x} $

    Figure: Function $ f (x) = x ^ 3 $ and the inverse function $ f ^ {- 1} (x) = \ sqrt [3] {x} $

    You can proceed in the same way for all other power functions that have an odd exponent. In the case of power functions that have an even exponent, you have to proceed differently, because two $ x $ values ​​are assigned to every $ y $ value except that of the vertex. For example, for the function $ y = x ^ 2 $, for the $ y $ value $ y = 4 $, both $ x = 2 $ and $ x = -2 $ are correct. Therefore, the definition range must be restricted.

    Let's look at the inverse function of the function $ f (x) = x ^ 2 $:

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    First of all, the definition set must be determined. We want to build the inverse function for all positive $ x $ values, $ x \ ge 0 $.

    1. Solve the function for $ x $:

    $ f (x) = x ^ 2 ~~~~~~~ | \ sqrt [2] {~~} $
    $ \ sqrt [2] {y} = x $

    2. Swap $ x $ and $ y $:

    $ f ^ {- 1} (x) = \ sqrt [2] {x} = \ sqrt {x} $, for all $ x \ ge 0 $.

    Figure: Function $ f (x) = x ^ 2 $ with inverse function $ f ^ {- 1} (x) = \ sqrt [2] {x} $

    With the tasks you can check your newly acquired knowledge. Good luck with it!

    Video: Simon Wirth

    Text: Chantal Rölle