Composite Functions
Let's see what happens when we combine functions.
a)
When $-1$ is the input, what is the final output?
b)
When $x$ is the input, what is the final output?
c)
When $x$ is the input, is the final output $f\left( g\left( x\right) \right)$ or $g\left(f\left(x\right)\right)$?
What happens when we reverse the order?
a)
When $-1$ is the input, what is the final output?
b)
When $x$ is the input, what is the final output?
c)
When $x$ is the input, is the final output $f\left( g\left( x\right) \right)$ or $g\left(f\left(x\right)\right)$?
If $f(x)=x-1$ and $g(x)=x^2-3x+4$ find:
a)
$f\left(g\left(5\right)\right)=$
*This is read "f of g of 5"
*This is read "f of g of 5"
b)
$\left(f\circ g\right)\left(5\right)=$
*This is also read "f of g of 5"
*This is also read "f of g of 5"
c)
$g\left(f\left(0\right)\right)=$
d)
$\left(g\circ f\right)\left(0\right)=$
e)
$f\left(f\left(1\right)\right)=$
f)
$g\left(g\left(1\right)\right)=$
g)
$f\left(g\left(x\right)\right)=$
*write your answer in the form $\square x^2\pm \square x\pm \square$
*write your answer in the form $\square x^2\pm \square x\pm \square$
h)
Since $f\left(g\left(x\right)\right)=x^2-3x+3$, we can now substitute directly. $f\left(g\left(5\right)\right)=$
i)
$g\left(f\left(x\right)\right)=$
*write your answer in the form $\square x^2\pm \square x\pm \square$
*write your answer in the form $\square x^2\pm \square x\pm \square$
j)
Since $g\left(f\left(x\right)\right)=x^2-5x+8$, we can now substitute directly. $g\left(f\left(0\right)\right)=$
k)
Solve for $x$ if $f\left(f\left(x\right)\right)=7$