# how do u do this function composition

• Jun 21st 2007, 08:23 PM
calchelp
how do u do this function composition
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)
• Jun 21st 2007, 08:38 PM
Jhevon
Quote:

Originally Posted by calchelp
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 $\left( f \cdot f^{-1} \right) (x) \mbox { or } \left( f \circ f^{-1} \right)(x) = f \left( f^{-1}(x) \right)$ ? 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)

$f(x) = y = x + 3$

For $f^{-1} (x)$ we switch $x$ and $y$, so we get:

$x = y + 3$

$\Rightarrow y = x - 3$

Therefore, $f^{-1}(x) = x - 3$

Now try to find $g^{-1} (x)$ and clarify the other questions. I'm pretty sure you mean $\left( f \circ f^{-1} \right)(x)$ though
• Jun 21st 2007, 10:21 PM
calchelp
sorry about that, i dont kno waht other way i could write it out

but yes you were right, i meant
http://www.mathhelpforum.com/math-he...708fbbe5-1.gif

thanks a lot!!!
• Jun 21st 2007, 10:26 PM
Jhevon
Quote:

Originally Posted by calchelp
sorry about that, i dont kno waht other way i could write it out

but yes you were right, i meant
http://www.mathhelpforum.com/math-he...708fbbe5-1.gif

thanks a lot!!!

Ok, i will do (b), the rest are similar.

Recall: $f(x) = x + 3$ and $f^{-1}(x) = x - 3$

$\Rightarrow \left(f \circ f^{-1} \right)(x) = f \left( f^{-1}(x) \right) = \left( f^{-1}(x) \right) + 3 = (x - 3) + 3 = x$

Note: It will always be the case that if $f(x)$ and $f^{-1}(x)$ represent functions that are the inverses of each other, then $\left( f \circ f^{-1} \right)(x) = \left( f^{-1} \circ f \right)(x) = x$

Now try the others and show your solutions. All I did was formed the composite function
• Jun 22nd 2007, 10:17 PM
calchelp
thanks a lot! i get it now