Learn how to find the inverse function of a one to one function ?f?. Solve inverse of a function examples and horizontal line test.
\(f(x_1)\neq f(x_2)\), whenever \(x_1\neq x_2\)
A x 1 2 3 4 5 6 7 f(x) 5.2 6.8 1.2 2.7 5.2 6.8 0
B x 1 2 3 4 5 6 7 f(x) 2.8 3.2 1.9 3.2 1.1 7 18
C x 1 2 3 4 5 6 7 f(x) 1.8 2 3.4 5.6 2.1 1.9 11
D x 1 2 3 4 5 6 7 f(x) 4.6 2.8 9.2 2.8 1 5 3.2
\(f^{-1}(y)=x\iff f(x)=y\)
for any \(y\) in \(B\).
\(f(2)=4 \Rightarrow\,f^{-1}(4)=2\)
\(f(3)=9 \Rightarrow\,f^{-1}(9)=3\)
\(f(4)=6 \Rightarrow\,f^{-1}(6)=4\)
or
\(f(f^{-1}(x))=x\)
e.g.
Step-1: \(f(x)=2x^3+3\)
\(\Rightarrow\,y=2x^3+3\)
\(\Rightarrow x^3=\dfrac {y-3}{2}\)
Step-2: \(\Rightarrow\,x=\left(\dfrac{y-3}{2}\right)^{1/3}\)
Step-3: \(y=\left(\dfrac{x-3}{2}\right)^{1/3}=f^{-1}(x)\)
A \(f^{-1}(x)=\dfrac{x+3}{2-5x}\)
B \(f^{-1}(x)=\dfrac{5x+1}{2x-3}\)
C \(f^{-1}(x)=sinx\)
D \(f^{-1}(x)=\dfrac{x^2+1}{2x^2+3}\)
The graph of \(f^{-1}\) is obtained by reflecting the graph of \(f\) about the line \(y=x\).
\(\left(f^{-1}\right)'(a)=\dfrac{1}{f'\left(f^{-1}(a)\right)}\:\:\:\:\:\:\:\rightarrow\,(1)\)
i.e. the derivative of \(f^{-1}\) at \(x=a\) is the reciprocal of the derivative of \(f\) at \(f^{-1}(a)\).
Suppose we are given an expression for a function \('f'\), to the derivative of \(f^{-1}\) at particular value either we need to find out the inverse function expression which is sometimes not possible or use the formula.
\(\left(f^{-1}\right)'(a)=\dfrac{1}{f'\left(f^{-1}(a)\right)}\)
A \(\dfrac{1}{2}\)
B \(\dfrac{1}{4}\)
C 4
D 2
If \((h,\,k)\) is on the graph of function \(f\) then, \(f(h)=k\) and \(f^{-1}(k)=h\).