if f(x) = x+3 and g(x) = 2x
a) f^ -1 (x)
b) [f * f^ -1](x)
c) [f^ -1 * f](x)
d) g^ -1 (x)
e) [g * g^ -1](x)
f) [g^ -1 * g](x)
does the "*" represent multiplication or a composite function?
That is, when you say [f * f^-1](x), do you mean $\displaystyle \left( f \cdot f^{-1} \right) (x) \mbox { or } \left( f \circ f^{-1} \right)(x) = f \left( f^{-1}(x) \right)$ ? Clarify what you want and we can continue
(a) and (d) are easy though. To get the inverse function, you switch the output and the input and solve for the output. That is, you switch x and y and solve for y.
(a)
$\displaystyle f(x) = y = x + 3$
For $\displaystyle f^{-1} (x)$ we switch $\displaystyle x$ and $\displaystyle y$, so we get:
$\displaystyle x = y + 3$
$\displaystyle \Rightarrow y = x - 3$
Therefore, $\displaystyle f^{-1}(x) = x - 3$
Now try to find $\displaystyle g^{-1} (x)$ and clarify the other questions. I'm pretty sure you mean $\displaystyle \left( f \circ f^{-1} \right)(x)$ though
Ok, i will do (b), the rest are similar.
Recall: $\displaystyle f(x) = x + 3$ and $\displaystyle f^{-1}(x) = x - 3$
$\displaystyle \Rightarrow \left(f \circ f^{-1} \right)(x) = f \left( f^{-1}(x) \right) = \left( f^{-1}(x) \right) + 3 = (x - 3) + 3 = x$
Note: It will always be the case that if $\displaystyle f(x)$ and $\displaystyle f^{-1}(x)$ represent functions that are the inverses of each other, then $\displaystyle \left( f \circ f^{-1} \right)(x) = \left( f^{-1} \circ f \right)(x) = x$
Now try the others and show your solutions. All I did was formed the composite function