Find the inverse of f(x)=7x informally.
Verify that f(f^-1(x))=x and f^-1(f(x))=x.
First of all, what is meant by informally???
Then, after I find the inverse informally, what do I do?
Please and Thank You's
As there isn't much to do to find it analytically, I suppose they mean guess at it?
Anyway, to find the inverse of a function f(x) use the following steps:
1) Set y = f(x)
2) Now let x = f(y) (This amounts to reflecting the graph over the line y = x, which graphically gives the inverse function.)
3) Solve for y. We'll have y = g(x) as the solution. g(x) is the inverse to f(x).
In this case:
1) Let y = f(x) = 7x
2) Let x = 7y
3) y = (1/7)x <-- This is the inverse function, $\displaystyle f^{-1}(x) = \frac{x}{7}$.
To prove this we need to show that $\displaystyle f(f^{-1}(x)) = f^{-1}(f(x)) = x$
So
$\displaystyle f(f^{-1}(x)) = f \left ( \frac{x}{7} \right ) = 7 \left ( \frac{x}{7} \right ) = x$ (Check!)
$\displaystyle f^{-1}(f(x)) = f^{-1}(7x) = \frac{(7x)}{7} = x$ (Check!)
-Dan