I face another problem childish but still a problem
Find the inverse functions of
a) x ---> 2x+3
b)x---> 3x-4
c) x---> 2(x+5)
d) x----> 1 (x-1)
I'm going to show you how to get the inverse function with example a). The remaining examples have to be done similarly.
to #a):
From $\displaystyle f:x\to2x+3$ you'll get the equation of the function f:
$\displaystyle f(x)=y=2x+3$
To find the equation of the inverse function you have to swap the variables and solve this equation for y:
$\displaystyle f:y=2x+3~\implies~f^{-1}:x=2y+3$
$\displaystyle x=2y+3~\implies~x-3=2y~\implies~\boxed{y=\frac12 x - \frac32}$
(Remark: You'll get the graph of $\displaystyle f^{-1}$ by reflection of the graph of f at the line y = x)
... and now it's your turn!
Another way of thinking about it: f(x)= 2x+ 3 tells you "whatever x is, first multiply by 2, then add 3". To find the inverse of f, do the opposite of each step, in the [b]opposite order. The opposite of "multiply by 2" is "divide by 2" and the opposite of "add 3" is "subtract 3". So the inverse of f is "subtract three, then divide by 2": $\displaystyle f^{-1}(x)= \frac{1}{2}(x- 3)= \frac{1}{2}x- \frac{3}{2}$ just as earboth got.