Inverse function facts for kids

Kids Encyclopedia Facts

An inverse function is a concept of mathematics. A function will calculate some output $y$ , given some input $x$ . This is usually written $f(x) = y$ . The inverse function does the reverse. Let's say $g$ is the inverse function of $f$ , then $g(y) = x$ . Or otherwise put, $g(f(x)) = x$ . An inverse function to $f$ is usually called $f^{-1}$ . It is not to be confused with $1/f$ , which is a reciprocal function.

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Examples

If $f(x) = x^3$ over real $x$ , then $f^{-1}(x) = \sqrt[3]{x}.$

To find the inverse function, swap the roles of $x$ and $y$ and solve for $y$ . For example, $y=e^x$ would turn to $x=e^y$ , and then $\ln x =y$ . This shows that the inverse function of $y=e^x$ is $y=\ln x$ .

Not all functions have inverse functions: for example, function $f(x) = |x|$ has none (because $|-1| = 1 = |1|$ , and $f^{-1}(x)$ cannot be both 1 and -1), but every binary relation has its own inverse relation.

In some cases, finding the inverse of a function can be very difficult to do.

Inverse element
Inverse triangle function
Inverse hyperbolic function
Invertible matrix

Images for kids

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

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Examples

Related pages

Images for kids

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