How Do You Convert Temperatures and Solve Inverse Functions?

[charlottecain]

Temperatures can be converted from Fahrenheit to Celsius using the
function f(x) = 5
/9
(x − 32).
(a) Calculate f(59).
(b) Find f
−1
(x), and verify that f
−1
(f(59)) = 59.
(c) Let K be the set {x : f(x) = x}. Find all elements of K and list K

 

[Evgeny.Makarov]

Hi, and welcome to the forum!

Temperatures can be converted from Fahrenheit to Celsius using the
function $f(x) = \frac59(x − 32)$.
(a) Calculate $f(59)$.

\[
f(59)=\frac59(59-32)=\frac59\cdot27=5\cdot\frac{27}{9}=5\cdot 3=15.
\]

(b) Find $f^{-1}(x)$, and verify that $f^{-1}(f(59)) = 59$.

To find the inverse of $f$ you need to solve the equation $y=\frac59(x − 32)$ for $x$, i.e., express $x$ through $y$. Can you do this? Start by multiplying both sides by $\frac95$.

(c) Let K be the set {x : f(x) = x}. Find all elements of K and list K

To do this you need to solve the equation $\frac59(x − 32)=x$. Can you do this?

For the future, please review the http://mathhelpboards.com/rules/, especially rule #11 (please click the "Expand" button in the top-left corner).