Inverse function Facts for Kids

An inverse function is a special idea in mathematics. Imagine you have a function like a machine. You put something in (an input), and it gives you something out (an output). We often write this as $f(x) = y$ , where $x$ is the input and $y$ is the output.

An inverse function does the exact opposite! It takes the output from the first function and turns it back into the original input. So, if $f$ takes $x$ and gives you $y$ , its inverse function (often called $f^{-1}$ ) takes $y$ and gives you $x$ back. This means Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f^{-1}(y) = x . It's like an "undo" button for the first function. Don't mix up $f^{-1}$ with $1/f$ , which is a different idea called a reciprocal function.

Understanding Inverse Functions
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Understanding Inverse Functions

What an Inverse Function Does

Think of a function as a rule that changes one number into another. For example, if a function doubles a number, its inverse function would halve that number. If you start with 5, double it to get 10, then halve 10, you get 5 back. This is how inverse functions work.

Finding an Inverse Function

To find an inverse function, you can follow a simple trick. Let's say you have a function written as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = f(x) .

First, swap the places of $x$ and $y$ .
Then, solve the new equation to get $y$ by itself again.

For example, if your function is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = x^3 (which means $y$ is $x$ multiplied by itself three times), to find its inverse:

Swap $x$ and $y$ : Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = y^3 .
Solve for $y$ : Take the cube root of both sides, so Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = \sqrt[3]{x} .

So, the inverse function of $f(x) = x^3$ is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f^{-1}(x) = \sqrt[3]{x} .

Another example: if Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = e^x (this is a special function involving the number e):

Swap $x$ and $y$ : Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x = e^y .
Solve for $y$ : Use the natural logarithm (written as $\ln$ ), which is the inverse of $e^x$ . So, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln x = y .

This shows that the inverse function of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = e^x is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = \ln x .

When Functions Don't Have Inverses

Not every function has a perfect inverse. For a function to have an inverse, each output must come from only one input. If two different inputs give the same output, the inverse function wouldn't know which input to go back to.

For example, consider the function $f(x) = |x|$ . This function gives you the positive value of any number.

If you put in -1, the output is 1 (Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): |-1| = 1 ).
If you put in 1, the output is also 1 (Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): |1| = 1 ).

If you tried to find an inverse for this, and you had the output 1, the inverse wouldn't know if it should give you -1 or 1 as the original input. Because of this, $f(x) = |x|$ does not have an inverse function.

Sometimes, finding the inverse of a function can be very tricky!

Inverse element
Inverse triangle function
Inverse hyperbolic function
Invertible matrix

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A function $f$ and its inverse $f -1$ . Because $f$ maps $a$ to 3, the inverse $f -1$ maps 3 back to $a$ .
If $f$ maps $X$ to $Y$ , then $f -1$ maps $Y$ back to $X$ .
The inverse of $g \circ f$ is $f -1 \circ g -1$ .
The graphs of $y = f (x)$ and $y = f -1(x)$ . The dotted line is $y = x$ .
The inverse of this cubic function has three branches.
The arcsine is a partial inverse of the sine function.

Inverse function facts for kids

Contents

Understanding Inverse Functions

What an Inverse Function Does

Finding an Inverse Function

When Functions Don't Have Inverses

Related pages

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See also