Answer to Question #118360 in Discrete Mathematics for inv

Question

1/ Find weather the two function are invertible or not, if it is find out its inverse (f^1(x))

1. f:[-π/2, π/2]→[-1,1]; f(x)=sin x
2. f:[0,π]→[-2,2]; f(x)=2cos x

2/ The function f(x)=5/9(x-32) converts Fahrenheit temperatures into Celsius, what is the opposite function for the opposite conversion?

Answer 1

1.1) Given function $f(x)=sin(x)$ is one-one in the given domain and $f(-\pi/2) = -1, f(\pi/2) = 1$ so given function is onto also.

Hence, given function is invertible.

Now, $f(x) = sin(x) \implies y = sin(x) \implies x = sin^{-1} (y)$.

So, Inverse function is $g(x) = sin^{-1}(x)$ , since $(fog)(x) = (gof)(x) = x$ .

1.2) Given function is one-one in the given domain and $f(0) = 2, f(\pi) = -2$, hence the given function is onto also. Hence given function is onto.

Now, $y = 2cos(x) \implies x = cos^{-1}(y/2)$.

So, Inverse function is $g(x) = cos^{-1}(x/2)$ since $(fog)(x) = (gof)(x) = x .$

2) Given $f(x) = (5/9)(x-32) \implies y = (5/9)(x-32)$

$\implies x = (\frac{9}{5}y)+32$.

Hence, Opposite function for the opposite conversion is $g(x) = \frac{9}{5} x +32.$