Inverse Functions

Inverse Functions

1.3

Inverse Functions

Detailed mind map of inverse function definition, notation, domain, and range.

Example:

Two identical circles representing an inverse function f^-1 mapping y to x.

Graph:

\(f\) has inverse function

Graph showing function with point on line, tested with horizontal line to check for inverse function.

\(h\) does not have inverse function

Graph showing function failing horizontal line test for inverse function, with line and point.

Properties:

Only one-to-one functions have inverse functions.
\(f\) and \(g\) are inverse functions of each other if and only if \(fg(x)=x\), \(x\) in domain of \(g\) and \(gf(x)=x\), \(x\) in domain of \(f\).
If \(f\) and \(g\) are inverse functions of each other, then
(a) domain of \(f=\) range of \(g\) and domain of \(g=\) range of \(f\).
(b) graph \(g\) is the reflection of graph \(f\) on the line \(y=x\).
Horizontal line test can be used to test the existence of inverse functions.
\(ff^{–1}(x)=f^{–1}f(x)=x\)

Function	Inverse Function
\(f(x)=x+1\)	\(f^{-1}(x)=x-1\)
\(f(x)=2x\)	\(f^{-1}(x)=\dfrac{1}{2}x\)
\(f(x)=2x+1\)	\(f^{-1}(x)=\dfrac{x-1}{2}\)
\(f(x)=\dfrac{x+1}{2}\)	\(f^{-1}(x)=2x-1\)

Inverse Functions

1.3

Inverse Functions

Detailed mind map of inverse function definition, notation, domain, and range.

Example:

Two identical circles representing an inverse function f^-1 mapping y to x.

Graph:

\(f\) has inverse function

Graph showing function with point on line, tested with horizontal line to check for inverse function.

\(h\) does not have inverse function

Graph showing function failing horizontal line test for inverse function, with line and point.

Properties:

Only one-to-one functions have inverse functions.
\(f\) and \(g\) are inverse functions of each other if and only if \(fg(x)=x\), \(x\) in domain of \(g\) and \(gf(x)=x\), \(x\) in domain of \(f\).
If \(f\) and \(g\) are inverse functions of each other, then
(a) domain of \(f=\) range of \(g\) and domain of \(g=\) range of \(f\).
(b) graph \(g\) is the reflection of graph \(f\) on the line \(y=x\).
Horizontal line test can be used to test the existence of inverse functions.
\(ff^{–1}(x)=f^{–1}f(x)=x\)

Function	Inverse Function
\(f(x)=x+1\)	\(f^{-1}(x)=x-1\)
\(f(x)=2x\)	\(f^{-1}(x)=\dfrac{1}{2}x\)
\(f(x)=2x+1\)	\(f^{-1}(x)=\dfrac{x-1}{2}\)
\(f(x)=\dfrac{x+1}{2}\)	\(f^{-1}(x)=2x-1\)