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

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.