Question

If \[f\left( x \right)=4-{{\left( x-7 \right)}^{3}}\], then find \[{{f}^{-1}}\left( x \right)\]

Answer 1

Hint: In this question, let us first assume the value of \[f\left( x \right)\] as some y. Now, substitute this in the given equation and find the value of x in terms of y. Then from the assumption we have that \[{{f}^{-1}}\left( y \right)\] as x. So, on further simplification we get the value of \[{{f}^{-1}}\left( x \right)\].

Complete step-by-step answer:
Now, from the given expression in the question we have
\[f\left( x \right)=4-{{\left( x-7 \right)}^{3}}\]
Inverse of a Function: Let \[f:A\to B\]is a bijective function, i.e. it is one-one and onto function. Then, we define \[g:B\to A\], such that \[f\left( x \right)=y\Rightarrow g\left( y \right)=x\], g is called the inverse of f and vice-versa. Symbolically, we write \[g={{f}^{-1}}\]
Thus, \[f\left( x \right)=y\Rightarrow {{f}^{-1}}\left( y \right)=x\]
Now, let us assume the given function in the question as y
\[\Rightarrow y=4-{{\left( x-7 \right)}^{3}}\]
Now, let us subtract 4 on both the sides
\[\Rightarrow y-4=-{{\left( x-7 \right)}^{3}}\]
Let us now multiply with -1 on both the sides to simplify it further
\[\Rightarrow 4-y={{\left( x-7 \right)}^{3}}\]
Now, let us apply the cube root on both the sides and simplify it further
\[\Rightarrow {{\left( 4-y \right)}^{\dfrac{1}{3}}}={{\left( {{\left( x-7 \right)}^{3}} \right)}^{\dfrac{1}{3}}}\]
Now, on further simplifying the above equation we get,
\[\Rightarrow {{\left( 4-y \right)}^{\dfrac{1}{3}}}=x-7\]
Let us now add 7 on both the sides and simplify it further
\[\Rightarrow {{\left( 4-y \right)}^{\dfrac{1}{3}}}+7=x\]
Now, from the assumed value we have
\[\Rightarrow f\left( x \right)=y\]
Now, on applying inverse on both sides to the above function we can write it as
\[\Rightarrow {{f}^{-1}}\left( y \right)=x\]
Now, let us substitute this condition obtained in the above equation
\[\Rightarrow {{\left( 4-y \right)}^{\dfrac{1}{3}}}+7={{f}^{-1}}\left( y \right)\]
Now, let us replace the variable y with x in the above obtained function
\[\therefore {{f}^{-1}}\left( x \right)={{\left( 4-x \right)}^{\dfrac{1}{3}}}+7\]

Note: Instead of assuming some variable to the given function and then simplifying it we can also solve it by directly finding the value of x from the given equation and then convert it accordingly to the required value. Both the methods give the same result.
It is important to note that while doing the arithmetic operations on both the sides to simplify we should not neglect any term or change the corresponding sign.