Question

Let \[f:R\to R\] be defined by \[f(x)=3x+4\], then \[{{f}^{-1}}(x)\] is
1. \[\dfrac{x+4}{3}\]
2. \[\dfrac{x-4}{3}\]
3. \[3x+4\]
4. Not defined

Answer 1

Hint: To find the function is one to one\[{{x}_{1}},{{x}_{2}}\in R\] such that \[f({{x}_{1}})=f({{x}_{2}})\]. If the function is invertible then it is one-one and onto then it is also known as a bijective function. To check if the function is invertible or not, first we have to check if the function is onto and one-one function.

Complete step by step answer:
A function is a relationship that states that each input should have just one output.
OR, to put it another way, a special type of connection (a set of ordered pairs) that follows a rule that is true in every case.
A function is a set of X-values that are linked to a set of Y-values. Onto function can be explained by considering two sets, Set A and Set B which consist of elements.
If there is at least one or more than one element matching with A for each element of B, the function is said to be onto or surjective.
A function is said to be bijective or bijection, if a function \[f:A\to B\] satisfies both injective (one-one function) and surjective function (onto function) properties. It means that for every element "b" in the codomain B, there is exactly one element "a" in the domain A. such that \[f(a)=b\]. One-one correspondence occurs when the function satisfies the requirement.
It is given that \[f:R\to R\] be defined by \[f(x)=3x+4\]
Before we finding the value of \[{{f}^{-1}}(x)\]
First we check whether a one-one function exists or not because function is invertible due to one-one and onto function.
\[f({{x}_{1}})=3{{x}_{1}}-4--(1)\]
\[f({{x}_{2}})=3{{x}_{2}}-4--(2)\]
By comparing equation \[(1)\] with equation \[(2)\]
\[3{{x}_{1}}-4=3{{x}_{2}}-4\]
By simplifying we get:
\[3{{x}_{1}}=3{{x}_{2}}\]
\[{{x}_{1}}={{x}_{2}}\]
\[\therefore f\] is one-one
We know one thing, if \[f(x)\] is invertible then \[f(x)\] is definitely a bijective function, which means \[f(x)\] will be one-one and onto.
Take the inverse of \[f(x)=3x-4\]
\[f(x)=y\]
By comparing this above two equation:
\[y=3x-4\]
After rearranging the term we get:
\[x=\dfrac{y+4}{3}\]
After taking the inverse of \[f(x)\] that is \[{{f}^{-1}}(x)\] you will get:
\[{{f}^{-1}}(x)=\dfrac{x+4}{3}\]

So, the correct answer is “Option 1”.

Note: Take a look at the question to see what is being asked. Your function concept should be obvious. A good assumption should be obvious. It is necessary to make a correct assumption. Do not make silly mistakes while substituting. Make sure you're equating it correctly so you don't get yourself mixed up.