The inverse of a square root function is a quadratic function where the domain is restricted.
In the graph, |\color{#333fb1}{f^{-1}(x)}| is the inverse of the square root function |\color{#333fb1}{f(x)}.| Note that the domain of |\color{#333fb1}{f^{-1}(x)}| is equivalent to the range of |\color{#333fb1}{f(x)}.| It's the same for |\color{#ec0000}{g^{-1}(x)}| and |\color{#ec0000}{g(x)}.|
So, just calculate the initial function’s range to find the domain restriction of the inverse function.
Use the following steps to find the inverse rule when the function rule is given.
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Switch |\color{#ec0000}{x}| and |\color{#333fb1}{y}| in the function rule.
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Isolate |\color{#333fb1}{y}.|
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Calculate the domain restriction of the inverse from the function’s range. For |f(x)=a\sqrt{b(x-h)}+k:|
|\text{dom}\ f^{-1}=\text{ran}\ f=\ ]-\infty,k]| if |a<0|
or
|\text{dom}\ f^{-1}=\text{ran}\ f=\ [k,+\infty[| if |a>0.| -
Write the rule of the inverse.
Find the rule of the inverse of the function |f(x)=2\sqrt{x-5}+10.|
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Switch |x| and |y| in the rule
||\begin{align}\color{#333fb1}{y}&=2\sqrt{\color{#ec0000}{x}-5}+10\\ \color{#ec0000}{x}&=2\sqrt{\color{#333fb1}{y}-5}+10\end{align}|| -
Isolate |y|
||\begin{align}x&=2\sqrt{y-5}+10\\ x-10&=2\sqrt{y-5}\\ \dfrac{x-10}{2}&=\sqrt{y-5}\\ \color{#ec0000}{\left(\color{black}{\dfrac{x-10}{2}}\right)^2}&=\color{#ec0000}{\left(\color{black}{\sqrt{y-5}}\ \right)^2}\\ \dfrac{(x-10)^2}{4}&=y-5\\ \dfrac{1}{4}(x-10)^2+5&=y\end{align}|| -
Calculate the restriction on the inverse’s domain
For |f(x),| |a=2| and |k=10.|
The parameter |a| of |f(x)| is positive, suggesting the function is defined above its vertex. So, the inverse function’s domain |f^{-1}(x),| which is equivalent to the range of |f(x),| is the following.
||\begin{align}\text{dom}\ f^{-1}=\text{ran}\ f&=[k,+\infty[\\&=[10,+\infty[\end{align}|| -
Give the rule
The function’s inverse |f(x)| is |f^{-1}(x)=\dfrac{1}{4}(x-10)^2+5| where |x\ge10.|
The curves of |\color{#333fb1}{f(x)}| and of |\color{#ec0000}{f^{-1}(x)}| from the previous example are found on the following Cartesian plane.
It is possible to plot the inverse of a function by interchanging the coordinates |x| and |y| of certain points. For example, the vertex |(\color{#333fb1}{5},\color{#333fb1}{10})| becomes the vertex |(\color{#ec0000}{10},\color{#ec0000}{5})| and the point |(\color{#333fb1}{9},\color{#333fb1}{14})| becomes |(\color{#ec0000}{14},\color{#ec0000}{9}).|
It can also be said that |\color{#ec0000}{f^{-1}(x)}| corresponds to the reflection of |\color{#333fb1}{f(x)}| regarding the line |y=x.| It is therefore possible to graph the inverse by reflection. For it to work, the scale of the |x|- and |y|-axes must have a ratio of |1:1.|