SOLUTION: 7. Let f(x) = to x squared - 6x + 9 ,g(x) = x - 3 find the following.
a.(f+g)(x) b.(f-g)(x) c.(fg)(x) d. (f/g)(x) e.(fog)(x)
f.(f-g)(5) g.g{f(x)} h. f
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Question 128321: 7. Let f(x) = to x squared - 6x + 9 ,g(x) = x - 3 find the following.
a.(f+g)(x) b.(f-g)(x) c.(fg)(x) d. (f/g)(x) e.(fog)(x)
f.(f-g)(5) g.g{f(x)} h. f{f(-3)}
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
(x)=f(x)+g(x))
Start with the given property
+g(x)=\left(x^2-6x+9\right)+\left(x-3\right))
Plug in and
+g(x)=x^2+\left(-6x+x\right)+\left(9-3\right))
Group like terms
+g(x)=x^2-5x+6)
Combine like terms
So (x)=x^2-5x+6)
------------------------------------------------------------------------
b)
(x)=f(x)-g(x))
Start with the given property
-g(x)=\left(x^2-6x+9\right)-\left(x-3\right))
Plug in and
-g(x)=x^2-6x+9-x+3)
Distribute the negative
-g(x)=x^2+\left(-6x-x\right)+\left(9+3\right))
Group like terms
-g(x)=x^2-7x+12)
Combine like terms
So (x)=x^2-7x+12)
------------------------------------------------------------------------
c)
(x)=f(x)\ast g(x))
Start with the given property
\ast g(x)=\left(x^2-6x+9\right)\ast\left(x-3\right))
Plug in and
\ast f(x)=\left(x-3\right)\ast\left(x^2-6x+9\right))
Rearrange the terms
\ast f(x)=x\left(x^2-6x+9\right)-3\left(x^2-6x+9\right))
Expand the expression. Remember something like expands to
\ast f(x)=x^3-6x^2+9x-3x^2+18x-27)
Distribute
\ast f(x)=x^3-9x^2+27x-27)
Combine like terms
So (x)=x^3-9x^2+27x-27)
------------------------------------------------------------------------
d)
(x)=\frac{f(x)}{g(x)})
Start with the given property
}{g(x)}=\frac{x^2-6x+9}{x-3})
Plug in and
}{g(x)}=\frac{\left(x-3\right)\left(x-3\right)}{x-3})
Factor to get
}{g(x)}=x-3)
Divide and simplify. Note:
So (x)=x-3)
------------------------------------------------------------------------
e)
(x)=f\left(g(x)\right))
Start with the given property
\right)=(x-3)^2-6(x-3)+9)
Plug in into each x of
\right)=x^2-6x+9-6(x-3)+9)
Foil
\right)=x^2-6x+9-6x+18+9)
Distribute
\right)=x^2-12x+36)
Combine like terms
So (x)=x^2-12x+36)
------------------------------------------------------------------------
f)
Start with the given property
Plug in and
-g(5)=\left(5^2-6(5)+9\right)-\left(5-3\right))
Plug in
-g(5)=\left(25-30+9\right)-\left(5-3\right))
Multiply
-g(5)=4-2)
Combine like terms
-g(5)=2)
Subtract
So (5)=2)
------------------------------------------------------------------------
g)
(x)=g\left(f(x)\right))
Start with the given property
\right)=(x^2-6x+9)-3)
Plug in into each x of
\right)=x^2-6x+6)
Combine like terms
So (x)=x^2-6x+6)
------------------------------------------------------------------------
h)
(x)=f\left(f(x)\right))
Start with the given property
\right)=(x^2-6x+9)^2-6(x^2-6x+9)+9)
Plug in into each x of
\right)=((-3)^2-6(-3)+9)^2-6((-3)^2-6(-3)+9)+9)
Now plug in
\right)=(9+18+9)^2-6(9+18+9)+9)
Multiply and square the given values
\right)=(36)^2-6(36)+9)
Add
\right)=1296-216+9)
Multiply and square the values
\right)=1089)
Combine like terms
So
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