Math Problem Statement
Perform the indicated operation for the given functions (composite functions).
Solution
Let's go through each problem one by one, solving the indicated operations.
1. g(x)=x3+2x, h(x)=x−2
Find (g∘h)(5):
- First, find h(5)=5−2=3.
- Now, find g(h(5))=g(3)=33+2(3)=27+6=33.
- Answer: (g∘h)(5)=33.
2. h(x)=2x+4, g(x)=−x+1
Find (h∘g)(−6):
- First, find g(−6)=−(−6)+1=6+1=7.
- Now, find h(g(−6))=h(7)=2(7)+4=14+4=18.
- Answer: (h∘g)(−6)=18.
3. g(x)=3x+3, f(x)=3x−4
Find g(f(−7)):
- First, find f(−7)=3(−7)−4=−21−4=−25.
- Now, find g(f(−7))=g(−25)=3(−25)+3=−75+3=−72.
- Answer: g(f(−7))=−72.
4. g(x)=−x3+5x2, f(x)=2x+4
Find g(f(−4)):
- First, find f(−4)=2(−4)+4=−8+4=−4.
- Now, find g(f(−4))=g(−4)=−(−4)3+5(−4)2=−(−64)+5(16)=64+80=144.
- Answer: g(f(−4))=144.
5. f(x)=x3−4, g(x)=4x+4
Find (f∘g)(0):
- First, find g(0)=4(0)+4=4.
- Now, find f(g(0))=f(4)=43−4=64−4=60.
- Answer: (f∘g)(0)=60.
6. g(n)=4n−5, f(n)=n3−5
Find g(f(3)):
- First, find f(3)=33−5=27−5=22.
- Now, find g(f(3))=g(22)=4(22)−5=88−5=83.
- Answer: g(f(3))=83.
7. g(x)=3x+2, h(x)=x+3
Find g(h(−7)):
- First, find h(−7)=−7+3=−4.
- Now, find g(h(−7))=g(−4)=3(−4)+2=−12+2=−10.
- Answer: g(h(−7))=−10.
8. h(t)=−t3−5t2, g(t)=−t−5
Find h(g(−5)):
- First, find g(−5)=−(−5)−5=5−5=0.
- Now, find h(g(−5))=h(0)=−(0)3−5(0)2=0.
- Answer: h(g(−5))=0.
9. f(t)=t3−5t, g(t)=3t−1
Find f(g(−1)):
- First, find g(−1)=3(−1)−1=−3−1=−4.
- Now, find f(g(−1))=f(−4)=(−4)3−5(−4)=−64+20=−44.
- Answer: f(g(−1))=−44.
10. f(x)=2x+3, g(x)=x2+5
Find (f∘g)(−4):
- First, find g(−4)=(−4)2+5=16+5=21.
- Now, find f(g(−4))=f(21)=2(21)+3=42+3=45.
- Answer: (f∘g)(−4)=45.
Let me know if you'd like more details on any of these problems or have any questions!
Related Questions:
- How do you find the domain of composite functions?
- What is the significance of function composition in real-life applications?
- Can every function be composed with another function?
- How can we reverse a composition of functions?
- What happens when you compose two linear functions?
Tip: Always evaluate the inner function first when performing function compositions.