Question

If f : R → R is defined by f(x) = 3x − 5, prove that f is a bijection and find its inverse

Answer 1

Given f(x) = 3x – 5

Let y = 3x – 5

y + 5 = 3x

⇒ `(y + 5)/3` = x

Let g(y) = `(y + 5)/3`

gof(x) = g(f(x))

= g(3x – 5)

= `(3x - 5 + 5)/3`

= `(3x)/5`

= x

gof(x) = x

fog(y) = f(g(y))

= `f((y + 5)/3)`

= `3((y + 5)/3) - 5`

= y + 5 – 5

fog(y) = y

∴ gof = I_x and fog = I_Y

Hence f and g are bijections and inverses to each other.

Hence f is a bijection and f^-1(y) = `(y + 5)/3`

Replacing y by x we get f^-1(x) = `(x + 5)/3`