Property 11
Inverse Function
Let \(f:A\to B\) be a one to one and onto function.
\(f^{-1}=\{(y,x)\mid(x,y)\in f\}\)
The \(f^{-1}\) function is defined as \(f^{-1}:\{(y,x):(x,y)\in f\}\)
\(f(x)=y\Rightarrow f^{-1}(y)=x\)
- \(f(x)=ax+b\quad\Rightarrow f^{-1}(x)=\samefrac{x-b}{a}\)
- \(f(x)=\samefrac{ax+b}{cx+d}\quad\Rightarrow f^{-1}(x)=\samefrac{-dx+b}{cx-a}\)
- \(f(x)=x\quad\Rightarrow f^{-1}(x)=x\)
- \(f(x)=\samefrac{k}{x}\quad\Rightarrow f^{-1}(x)=\samefrac{k}{x}\)
- \(f(x)=k-x\quad\Rightarrow f^{-1}(x)=k-x\)
1.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=3x+1\)
\(\Rightarrow f^{-1}(x)=?\)
2.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=4x-6\)
\(\Rightarrow f^{-1}(x)=?\)
3.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=\samefrac{x-2}{3}\)
\(\Rightarrow f^{-1}(x)=?\)
4.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x)=\samefrac{4x+2}{3}\)
\(\Rightarrow f^{-1}(x)=?\)
5.
\(f:\mathbb{R}\setminus\{2\}\to \mathbb{R}\setminus\{4\}\)
\(f(x)=\samefrac{4x-6}{x-2}\)
\(\Rightarrow f^{-1}(x)=?\)
6.
\(f:\mathbb{R}\setminus\{1\}\to \mathbb{R}\setminus\{-2\}\)
\(f(x)=\samefrac{-2x+5}{x-1}\)
\(\Rightarrow f^{-1}(x)=?\)
7.
\(f:\mathbb{R}\setminus\{1\}\to \mathbb{R}\setminus\{2\}\)
\(f(x)=\samefrac{5-2x}{1-x}\)
\(\Rightarrow f^{-1}(x)=?\)
8.
\(f:\mathbb{R}\setminus\{3\}\to \mathbb{R}\setminus\{1\}\)
\(f\left(\samefrac{3x+4}{x-1}\right)=x\)
\(\Rightarrow f^{-1}(x)=?\)
9.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(2x+1)=3x\)
\(\Rightarrow f^{-1}(6)=?\)
10.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(x-5)=3x+1\)
\(f^{-1}(7)=?\)
11.
\(f:\mathbb{R}\to \mathbb{R}\)
\(f(3x+1)=\samefrac{x+1}{3}\)
\(f^{-1}(2)=k\)
\(\Rightarrow k=?\)
12.
\(f(x)=\begin{cases}x^2-1&x>2\\x+1&x\le2\end{cases}\)
\(\Rightarrow f^{-1}(8)+f^{-1}(0)=?\)
13.
\(f:[2,\infty)\to [1,\infty)\)
\(f(x)=x^2-4x+5\)
\(\Rightarrow f^{-1}(x)=?\)