Question

The formula for converting from Fahrenheit to Celsius temperatures is y = `(5x)/9 - 160/9`. Find the inverse of this function and determine whether the inverse is also a function

Answer 1

Let f(x) = `(5x)/9 - 160/9`

= `(5x - 160)/9`

Given y = `(5x)/9 - 160/9`

= `(5x - 160)/9`

Then 9y = 5x – 160

⇒ 5x = 9y + 160

⇒ x = `(9y + 160)/5`

Let g(y) = `(9y + 160)/5`

g o f(x) = g(f(x))

= `g((5x - 160)/9)`

= `(9((5x - 160)/9) +160)/5`

= `(5x - 160 + 60)/5`

= `(5x)/5`

g o f(x) = x

f o g(y) = f(g(y))

= `f((9y + 160)/5)`

= `(5((9y + 160)/5) - 160)/9`

= `(9y + 160 - 160)/9`

= `(9y)/9`

f o g(y) = y

Thus g o f =I_x nd f o g = I_y

This shows that f and g are bijections and inverses of each other.

f^–1(y = `(9y + 160)/5`

Replacing y by x we get f^–1(x) = `(9x + 160)/5`

= `(9x)/5 + 32`