Example
Let
f(x)=2x3−5x+3 and
h(x)=x−5,
Find the following:
(f+h)(x)[/latex]and[latex](h−f)(x)
Answer:
(f+h)(x)=f(x)+h(x)=(2x3−5x+3)+(x−5)=2x3−5x+3+x−5=2x3−4x−2combine like termssimplify
(h−f)(x)=h(x)−f(x)=(x−5)−(2x3−5x+3)=x−5−2x3+5x−3=−2x3+6x−8combine like termssimplify
In our next example, we will evaluate a sum of functions and show that you can get to the same result in one of two ways.
Example
Let
f(x)=2x3−5x+3 and
h(x)=x−5
Evaluate:
(f+h)(2)
Show that you get the same result by
1) Evaluating the functions first then performing the indicated operation on the result.
2) Performing the operation on the functions first then evaluating the result.
Answer:
1) First, we will evaluate the functions separately:
f(2)=2(2)3−5(2)+3=16−10+3=9
h(2)=(2)−5=−3
Now we will perform the indicated operation using the results:
(f+h)(2)=f(2)+h(2)=9+(−3)=6
2) We can get the same result by adding the functions first and then evaluating the result at x=2.
(f+h)(x)=f(x)+h(x)=2x3−4x−2 from the previous example.
Now we can evaluate this result at x=2
(f+h)(2)=2(2)3−4(2)−2=16−8−2=6
Both methods give the same result, and both require about the same amount of work.
Example
Let
g(t)=2t3−t2+7 and
f(t)=5t2−3
Find
(g⋅f)(t), and evaluate
(g⋅f)(−1)
Answer:
(g⋅f)(t)=(2t3−t2+7)(5t2−3)=(2t3⋅(5t2)−t2⋅(5t2)+7⋅(5t2))+(2t3⋅(−3)−t2⋅(−3)+7⋅(−3))apply the distributive property=(10t5−5t4+35t2)+(−6t3+3t2−21)simplify=10t5−5t4−6t3+38t2−21combine like terms
Evaluate
(g⋅f)(−1)
(g⋅f)(t)=10t5−5t4−6t3+38t2−21(g⋅f)(−1)=10(−1)5−5(−1)4−6(−1)3+38(−1)2−21=−10−5+6+38−21=8
In the next example, we will divide polynomial functions and then evaluate the new function.
Example
Given
p(x)=2x2+x−15 and
r(x)=x+3
Find
rp(x) and evaluate
rp(2)
Answer:
We can use synthetic division for this polynomial division since the coefficient on r(x)=x+3 is 1.
This result means that rp(x)=x+32x2+x−15=2x−5
Now evaluate this quotient for x=−3 both ways as we did in a previous example.
First, we will evaluate the result after polynomial division:
rp(x)=2x−5rp(2)=2(2)−5=4−5=−1
Next, we will evaluate each function for x=2, then we will divide the results.
p(2)=2(2)2+(2)−15=8+2−15=−5
r(2)=(2)+3=5
Divide the results:
rp(2)=5−5=−1
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