Question

What is the inverse of $y={{x}^{\dfrac{1}{5}}}+1$?

Answer 1

Hint: We know that to find the inverse of any function $f\left( x \right)=y$, we need to solve this for $x$ to get a function of the form $h\left( y \right)=x$. We can then interchange the variables x and y, to get our required inverse function.

Complete step by step solution:
We know that if any function f takes x to y, then the inverse of function f will take y to x. We are very well aware that the inverse of any function $f$, is denoted as ${{f}^{-1}}$.
We can also define the inverse of a function as, if $f\left( x \right)=y$ and $h\left( y \right)=x$, then f and h are inverse functions of one another.
We must remember that to calculate the inverse of any function $f\left( x \right)=y$, we need to solve this equation and accordingly find the equation for variable $x$. Then, we will have a function of the form $h\left( y \right)=x$ and we can just interchange the variables x and y, to get the required inverse function.
In our question, we have $y={{x}^{\dfrac{1}{5}}}+1$. Let us solve this equation and find $x$.
Subtracting 1 on both sides, we get
$y-1={{x}^{\dfrac{1}{5}}}+1-1$
We can write this as,
$y-1={{x}^{\dfrac{1}{5}}}$
Let us now raise both sides by the power of 5. Thus, we get
${{\left( y-1 \right)}^{5}}={{\left( {{x}^{\dfrac{1}{5}}} \right)}^{5}}$
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$. So, by using this property, we get
${{\left( y-1 \right)}^{5}}={{x}^{\dfrac{1}{5}\times 5}}$
Hence, we now have
${{\left( y-1 \right)}^{5}}=x$
We can rewrite this as
$x={{\left( y-1 \right)}^{5}}$
Now, we need to interchange the variables x and y to get the answer in standard form. So, we now have
$y={{\left( x-1 \right)}^{5}}$
Thus, for $f\left( x \right)={{x}^{\dfrac{1}{5}}}+1$, the inverse function is ${{f}^{-1}}\left( x \right)={{\left( x-1 \right)}^{5}}$.

Note: We must remember that not every function has an inverse function. An inverse function exists only for invertible functions. We must keep in mind that though the symbol of inverse functions seems as if it is reciprocated, that is not the case. Inverse and reciprocal are two different things.