Question

consider f:R - R given by f(x) = 5x+2.
a.) show that f is one-one
b.) Is f invertible? Justify your answer.​

Answer 1

Solution :

Given function :

f : R → R , f(x) = 5x + 2

★ f(x) is one-one :

Let f(x1) = f(x2)

=> 5x1 + 2 = 5x2 + 2

=> 5x1 = 5x2

=> x1 = x2

→ Whenever f(x1) = f(x2) , then x1 = x2 .

Whenever f(x1) = f(x2) , then x1 = x2 . Thus , f(x) is one-one .

★ Whether f(x) is invertible :

→ If a function is one-one onto , then f(x) is invertible .

• Here , f(x) is one-one (shown above) .

Now ,

Let y = f(x)

=> y = 5x + 2

=> 5x = y - 2

=> x = (y - 2)/5

=> For x to be real , y can be any real number .

=> Range = R

=> Range = Co-domain

=> f(x) is onto .

→ Since f(x) is one-one onto ,

thus it is invertible function .