Question

How do you find the inverse of $f(x) = {x^5} + {x^3} + x$ ?

Answer 1

A function is defined as an expression containing two variable terms; one term is called the independent variable as its value doesn’t depend on the value of the other variable, the other term is called the dependent variable as it changes with the value of the independent variable. In this question, we let the function equal to be “y” (dependent variable). Now, we will express x in terms of y, such that x is now the dependent variable and y is the independent variable. This way we can find the inverse of the given function.

Complete step by step answer:
The inverse of a function can simply be defined as the reflection of the given function in the line $y = x$ , this line passes through the origin and divides a graph exactly into two parts.The given function is not linear and involves high powers of x.

So it cannot be solved by simply expressing x in terms of y. In such cases, we cannot find the exact expression for writing the inverse of the function but for finding an approximate value, we can use newton’s method of finding the approximate values of functions of a higher degree.

We are given that $f(x) = {x^5} + {x^3} + x$
We can find whether the inverse of this function exists or not as follows –
Differentiating this function, we get –
$f'(x) = 5{x^4} + 3{x^2} + 1$
For all the values of ${x^4} \geqslant 0\,and\,{x^2} \geqslant 0$
$\Rightarrow f'(x) \geqslant 1$
So, the function is continuous and monotonically increasing and thus has an inverse.

Hence the function can be defined as: ${f^{ - 1}}(y) = x \in R:{x^5} + {x^3} + x = y$

Note: An inverse function or an anti function is defined as a function, which can reverse into another function. In simple words, if any function “f” takes x to y then, the inverse of “f” will take y to x. If the function is denoted by ‘f’ or ‘F’, then the inverse function is denoted by ${f^{-1}}$.