Question

Show that f: R → 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

it is given that a function f : R ---> R is given by f(x) = 3x - 4.

let's take $x_1$ and $x_2$ in domain of given function such that $f(x_1)=f(x_2)$

or, $3x_1-4=3x_2-4$

or, $3x_1=3x_2$

or, $x_1=x_2$

hence, f is one - one function .

here given function , f(x) = 3x - 4 is a linear polynomial function. we know, every polynomial function is defined in all real numbers and range of polynomial function also belongs to all real numbers.
e.g., Range $\in$ R
given, co-domain $\in$ R

here, co-domain = Range
so, f is onto function.

now, f(x) = y = 3x - 4

or, y + 4 = 3x

or, x = (y + 4)/3

hence, $f^{-1}(x)=\frac{x+4}{3}$

f⁻¹ (9) = (9 + 4)/3 = 13/3

f⁻¹ (-2) = (-2 + 4)/3 = 2/3

Answer 2

Answer:

Step-by-step explanation:

it is given that a function f : R ---> R is given by f(x) = 3x - 4.

let's take  and  in domain of given function such that  

or,  

or,  

or,  

hence, f is one - one function .

here given function , f(x) = 3x - 4 is a linear polynomial function. we know, every polynomial function is defined in all real numbers and range of polynomial function also belongs to all real numbers.

e.g., Range  R

given, co-domain  R

here, co-domain = Range

so, f is onto function.

now, f(x) = y = 3x - 4

or, y + 4 = 3x

or, x = (y + 4)/3

hence,  

f⁻¹ (9) = (9 + 4)/3 = 13/3

f⁻¹ (-2) = (-2 + 4)/3 = 2/3