
Let $f:R\to R:f\left( x \right)=10x+3$. Find ${{f}^{-1}}$.
Answer
444.9k+ views
Hint: In this problem we have given the function $f\left( x \right)$ as $10x+3$, and asked to calculate the value of ${{f}^{-1}}$. For this we will first take the assumption $y=10x+3$ and solve for the $x$. To solve the above equation, we will apply the reverse athematic operations for the operations we have in the equation. We can observe that the addition and multiplication operations are involved in the given equation. So, we will apply reverse arithmetic operations for addition and multiplication which are subtraction and division. After applying all the arithmetic operations, we will get the value of $x$ which is our required value.
Complete step by step solution:
Given that, $f:R\to R:f\left( x \right)=10x+3$.
Let us assume $y=10x+3$. To find the value of ${{f}^{-1}}$ we need to have the value of $x$ from the above equation.
In the above equation we can observe that $3$ is in addition, so we will subtract $3$ from both sides of the above equation, then we will get
$\Rightarrow y-3=10x+3-3$
We know that $+a-a=0$, then the above equation is modified as
$\Rightarrow y-3=10x$
In the above equation we can observe that $10$ is in multiplication, so we will divide with $10$ on both sides of the above equation, then we will get
$\Rightarrow \dfrac{y-3}{10}=\dfrac{10x}{10}$
We know that $\dfrac{a}{a}=1$, then we will have
$\Rightarrow \dfrac{y-3}{10}=x$
Here we have the value of $x$ as $\dfrac{y-3}{10}$. Hence the value of ${{f}^{'}}\left( x \right)$ is $\dfrac{x-3}{10}$.
Note: We can check whether the obtained answer is correct or wrong by using the relation between the function $f\left( x \right)$, ${{f}^{'}}\left( x \right)$ which is “if $f\left( b \right)=a$ then ${{f}^{-1}}\left( a \right)$ must be $b$”. We can also have the relation ${{f}^{-1}}\left( f\left( a \right) \right)=a$. From these relations we can check if the result is correct or wrong.
Complete step by step solution:
Given that, $f:R\to R:f\left( x \right)=10x+3$.
Let us assume $y=10x+3$. To find the value of ${{f}^{-1}}$ we need to have the value of $x$ from the above equation.
In the above equation we can observe that $3$ is in addition, so we will subtract $3$ from both sides of the above equation, then we will get
$\Rightarrow y-3=10x+3-3$
We know that $+a-a=0$, then the above equation is modified as
$\Rightarrow y-3=10x$
In the above equation we can observe that $10$ is in multiplication, so we will divide with $10$ on both sides of the above equation, then we will get
$\Rightarrow \dfrac{y-3}{10}=\dfrac{10x}{10}$
We know that $\dfrac{a}{a}=1$, then we will have
$\Rightarrow \dfrac{y-3}{10}=x$
Here we have the value of $x$ as $\dfrac{y-3}{10}$. Hence the value of ${{f}^{'}}\left( x \right)$ is $\dfrac{x-3}{10}$.
Note: We can check whether the obtained answer is correct or wrong by using the relation between the function $f\left( x \right)$, ${{f}^{'}}\left( x \right)$ which is “if $f\left( b \right)=a$ then ${{f}^{-1}}\left( a \right)$ must be $b$”. We can also have the relation ${{f}^{-1}}\left( f\left( a \right) \right)=a$. From these relations we can check if the result is correct or wrong.
Recently Updated Pages
Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Master Class 12 Physics: Engaging Questions & Answers for Success

Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Trending doubts
What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

What are the major means of transport Explain each class 12 social science CBSE

Draw a neat and well labeled diagram of TS of ovary class 12 biology CBSE

When was the first election held in India a 194748 class 12 sst CBSE

What is pseudocoelom Where is it located class 12 biology CBSE

State the postulates of special theory of relativi class 12 physics CBSE
