Inverse Functions #1

Method 1

Let $y$ be the output of $f(x)$ . The inverse function should therefore take in a $y$ value and return the corresponding $x$ value. Since $\displaystyle y={{x-3}\over{2x+5}}$ , and we want to find $x$ in terms of $y$ , we just need to change the subject of the function from $y$ to $x$ .

1. $y(2x+5)=x-3$

2. $2xy+5y=x-3$

3. $2xy-x=-3-5y$

4. $x(2y-1)=-3-5y$

5. $\displaystyle x={{-3-5y}\over{2y-1}}$

Therefore $\displaystyle f^{-1}(y)={{-3-5y}\over{2y-1}}$ . We now need to evaluate $f^{-1}(1)$ , so we substitute $y=1$ into $f^{-1}(y)$ .

$\displaystyle{{-3-5(1)}\over{2(1)-1}}$

$\displaystyle{{{-8}\over{1}}=\boxed{-8}}$

Method 2

$f^{-1}(1)$ is equivalent to the output of $f(x)$ , to which the answer is $1$ . This means, we can substitute the output as 1 in the equation for function $f$ .

$\displaystyle 1={{x-3}\over{2x+5}}$

$2x+5=x-3$

$x = \boxed{-8}$