Question

If \[f(x) = 1 - 4x\] and \[{f^{ - 1}}(x)\] is the inverse of \[f(x)\], then value of \[f( - 3){f^{ - 1}}( - 3) = \]is,
A) \[1\]
B) \[3\]
C) \[4\]
D) \[10\]
E) \[13\]

Answer 1

Hint- In Mathematics, if the function \[f\] applied to an input \[x\] and gives a result as \[y\], then applying its inverse function \[g\] to \[y\] gives the result \[x\], and vice versa that is \[f(x) = y\]if and only if \[g(y) = x\].
The inverse function is also called the reverse function or anti function.
Value of a function:
Let us consider a function \[f(x) = a{x^2} + bx + c\], then to find the value of the function at \[x = \alpha \] we simply replace all the \[x\]by \[\alpha \]. And hence, we get,
\[f(\alpha ) = a{\alpha ^2} + b\alpha + c\].

Complete step by step answer:
It is given that,
\[f(x) = 1 - 4x\]
At first, we will find the inverse of the function \[f(x)\].
Let us consider, \[f(x) = y\]
As per the problem we can write as,
\[y = 1 - 4x\]
To find the inverse function we will represent \[x\]in terms of \[y\].
So, simplifying we get,
\[x = \dfrac{{1 - y}}{4}\]
So, the inverse function of \[f(x)\] is \[{f^{ - 1}}(x) = \dfrac{{1 - x}}{4}\].
Now we will find the value of \[f(x)\] at \[x = - 3\].
Substitute the value of \[x = - 3\] in \[f(x)\] we get,
\[f( - 3) = 1 - 4( - 3)\]
Simplifying we get,
\[f( - 3) = 13\]
Similarly, we will find the value of \[{f^{ - 1}}(x)\] at \[x = - 3\].
Substitute the value of \[x = - 3\] in \[{f^{ - 1}}(x)\] we get,
\[{f^{ - 1}}( - 3) = \dfrac{{1 - ( - 3)}}{4}\]
Simplifying we get,
\[{f^{ - 1}}( - 3) = 1\]
So, \[f( - 3){f^{ - 1}}( - 3) = 13 \times 1 = 13\]
Hence, the correct option is (E) \[13\]

Note – A function works like a machine. It has input and an output. Output is related to the input on the basis of the relation. Again, the inverse of the function is just the reverse. The main parts of the function are: input, relation and output.
Here, the output of the function is the input and gives input as the output.