Question

Show that \[f\,:\,R\to R\] given by f(x) \[=3x-4\] is one–one and onto. Find its inverse function.
Also find \[{{f}^{1}}(9)\,\,\,and\,\,\,{{f}^{1}}(2)\]

Answer 1

Hint: Use the properties of one-one function to solve the problem. In a one-one function every element of a domain has a unique image in its co-domain. Use the properties of onto function, in onto function the range equal to co-domain of the function. Use the definition of domain and range, for domain we solve for real values of \[x\] and for range we solve for real values of \[y.\]

Complete step-by-step answer:
One–One: A function
\[f:A\to B\]
is said to be one-one function if different element of A have different image in B. one-one function is also called injective function
Onto function (surjective function):
If range of the function equal to the co-domain of the function then function is known as onto or surjective.
Given that
\[f(x)\,=3x-4\]
"Domain and range of the function
\[f(x)\,=3x-4\]
 is a real number because function is polynomial.
Let
\[{{x}_{1}}\]
 and
\[{{x}_{2}}\]
 in the domain of the function
\[f(x)\,=3x-4\]
Therefore
Condition of the one-one function
\[f({{x}_{1}})=f({{x}_{2}})\]
Then
\[\Rightarrow 3{{x}_{1}}-4=3{{x}_{2}}-4\]
Simplify the expression
\[\Rightarrow 3{{x}_{1}}=3{{x}_{2}}\]
Cancel out \[3\]from both sides
\[\Rightarrow {{x}_{1}}={{x}_{2}}\]
So function is one-one function
Condition for the onto function
Range = co-domain
Then
Simplify the expression
\[y+4=3x\]
Rewrite the equation after simplification
\[\Rightarrow 3x=y+4\]
Then
\[\Rightarrow x=\dfrac{y+4}{3}\]
Therefore
\[\Rightarrow {{f}^{1}}(x)=\dfrac{x+4}{3}\]
Therefore
\[\Rightarrow {{f}^{1}}(9)=\dfrac{9+4}{3}\]
\[\Rightarrow {{f}^{1}}(9)=\dfrac{13}{3}\]
By putting \[x=-2,\] we get
\[{{f}^{1}}(-2)=\dfrac{-2+4}{3}\]
Simply it
\[\Rightarrow {{f}^{1}}(-2)=\dfrac{2}{3}\]

Note: This problem is also solved with the concept of the graphical transformation. We can also find the domain and range by the graph of the function for the values on x-axis and y-axis.
In one-one function the graph of the function is cut by a horizontal line parallel to the x-axis at a single point only but in case of many one function the graph of the function cuts more than one point by the horizontal line.
In case of on-to function the range of function is equal to the co-domain of the function so the graph of the function represents the value of range that also lies in the co-domain of the function.