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Question: How do you find \[{{f}^{-1}}\left( x \right)\] given \[f\left( x \right)={{x}^{2}}-4x+3\]?...

How do you find f−1(x){{f}^{-1}}\left( x \right) given f(x)=x2−4x+3f\left( x \right)={{x}^{2}}-4x+3?

Explanation

Solution

In this problem, we have to find the inverse function for the given function. We know that the inverse function is a function that reverses another function. We can first write the given equation then we can replace f(x) with y. We can then replace y with x and x with y. We will get a new equation by switching the x-y values. we can solve for y and we can replace y with f−1(x){{f}^{-1}}\left( x \right) to get the inverse function.

Complete step by step solution:
We know that the given function is,
f(x)=x2−4x+3f\left( x \right)={{x}^{2}}-4x+3……. (1)
We have to find f−1(x){{f}^{-1}}\left( x \right) for the given function.
We know that the inverse function is a function that reverses another function. If the function applied to the input x gives a result of y then applying its inverse function to y gives the result of x.
We can now replace f(x) with y in the function (1),
⇒y=x2−4x+3\Rightarrow y={{x}^{2}}-4x+3
We can now replace y with x and x with y. We will get a new equation by switching the x-y values.
⇒x=y2−4y+3\Rightarrow x={{y}^{2}}-4y+3
We can now write the above equation as,

& \Rightarrow x=\left( {{y}^{2}}-4x+4-4 \right)+3 \\\ & \Rightarrow x={{\left( y-2 \right)}^{2}}-4+3 \\\ & \Rightarrow x={{\left( y-2 \right)}^{2}}-1 \\\ \end{aligned}$$ We can now solve for y by taking square root on both sides, we get $$\begin{aligned} & \Rightarrow x+1={{\left( y-2 \right)}^{2}} \\\ & \Rightarrow y-2=\pm \sqrt{x+1} \\\ & \Rightarrow y=2\pm \sqrt{x+1} \\\ \end{aligned}$$ Now we can replace y with $${{f}^{-1}}\left( x \right)$$, we get $$\Rightarrow {{f}^{-1}}\left( x \right)=2\pm \sqrt{x+1}$$ Therefore, the inverse function, $${{f}^{-1}}\left( x \right)=2\pm \sqrt{x+1}$$. **Note:** Students make mistakes while understanding the exact concept of the inverse function. We should always remember that an inverse function is reverse over another function, If the function applied to the input x gives a result of y then applying its inverse function to y gives the result of x.