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+2xg(x) = x^3 + 2x, h(x)=x2h(x) = x - 2

Find (gh)(5)(g \circ h)(5):

  • First, find h(5)=52=3h(5) = 5 - 2 = 3.
  • Now, find g(h(5))=g(3)=33+2(3)=27+6=33g(h(5)) = g(3) = 3^3 + 2(3) = 27 + 6 = 33.
  • Answer: (gh)(5)=33(g \circ h)(5) = 33.

2. h(x)=2x+4h(x) = 2x + 4, g(x)=x+1g(x) = -x + 1

Find (hg)(6)(h \circ g)(-6):

  • First, find g(6)=(6)+1=6+1=7g(-6) = -(-6) + 1 = 6 + 1 = 7.
  • Now, find h(g(6))=h(7)=2(7)+4=14+4=18h(g(-6)) = h(7) = 2(7) + 4 = 14 + 4 = 18.
  • Answer: (hg)(6)=18(h \circ g)(-6) = 18.

3. g(x)=3x+3g(x) = 3x + 3, f(x)=3x4f(x) = 3x - 4

Find g(f(7))g(f(-7)):

  • First, find f(7)=3(7)4=214=25f(-7) = 3(-7) - 4 = -21 - 4 = -25.
  • Now, find g(f(7))=g(25)=3(25)+3=75+3=72g(f(-7)) = g(-25) = 3(-25) + 3 = -75 + 3 = -72.
  • Answer: g(f(7))=72g(f(-7)) = -72.

4. g(x)=x3+5x2g(x) = -x^3 + 5x^2, f(x)=2x+4f(x) = 2x + 4

Find g(f(4))g(f(-4)):

  • First, find f(4)=2(4)+4=8+4=4f(-4) = 2(-4) + 4 = -8 + 4 = -4.
  • Now, find g(f(4))=g(4)=(4)3+5(4)2=(64)+5(16)=64+80=144g(f(-4)) = g(-4) = -(-4)^3 + 5(-4)^2 = -(-64) + 5(16) = 64 + 80 = 144.
  • Answer: g(f(4))=144g(f(-4)) = 144.

5. f(x)=x34f(x) = x^3 - 4, g(x)=4x+4g(x) = 4x + 4

Find (fg)(0)(f \circ g)(0):

  • First, find g(0)=4(0)+4=4g(0) = 4(0) + 4 = 4.
  • Now, find f(g(0))=f(4)=434=644=60f(g(0)) = f(4) = 4^3 - 4 = 64 - 4 = 60.
  • Answer: (fg)(0)=60(f \circ g)(0) = 60.

6. g(n)=4n5g(n) = 4n - 5, f(n)=n35f(n) = n^3 - 5

Find g(f(3))g(f(3)):

  • First, find f(3)=335=275=22f(3) = 3^3 - 5 = 27 - 5 = 22.
  • Now, find g(f(3))=g(22)=4(22)5=885=83g(f(3)) = g(22) = 4(22) - 5 = 88 - 5 = 83.
  • Answer: g(f(3))=83g(f(3)) = 83.

7. g(x)=3x+2g(x) = 3x + 2, h(x)=x+3h(x) = x + 3

Find g(h(7))g(h(-7)):

  • First, find h(7)=7+3=4h(-7) = -7 + 3 = -4.
  • Now, find g(h(7))=g(4)=3(4)+2=12+2=10g(h(-7)) = g(-4) = 3(-4) + 2 = -12 + 2 = -10.
  • Answer: g(h(7))=10g(h(-7)) = -10.

8. h(t)=t35t2h(t) = -t^3 - 5t^2, g(t)=t5g(t) = -t - 5

Find h(g(5))h(g(-5)):

  • First, find g(5)=(5)5=55=0g(-5) = -(-5) - 5 = 5 - 5 = 0.
  • Now, find h(g(5))=h(0)=(0)35(0)2=0h(g(-5)) = h(0) = -(0)^3 - 5(0)^2 = 0.
  • Answer: h(g(5))=0h(g(-5)) = 0.

9. f(t)=t35tf(t) = t^3 - 5t, g(t)=3t1g(t) = 3t - 1

Find f(g(1))f(g(-1)):

  • First, find g(1)=3(1)1=31=4g(-1) = 3(-1) - 1 = -3 - 1 = -4.
  • Now, find f(g(1))=f(4)=(4)35(4)=64+20=44f(g(-1)) = f(-4) = (-4)^3 - 5(-4) = -64 + 20 = -44.
  • Answer: f(g(1))=44f(g(-1)) = -44.

10. f(x)=2x+3f(x) = 2x + 3, g(x)=x2+5g(x) = x^2 + 5

Find (fg)(4)(f \circ g)(-4):

  • First, find g(4)=(4)2+5=16+5=21g(-4) = (-4)^2 + 5 = 16 + 5 = 21.
  • Now, find f(g(4))=f(21)=2(21)+3=42+3=45f(g(-4)) = f(21) = 2(21) + 3 = 42 + 3 = 45.
  • Answer: (fg)(4)=45(f \circ g)(-4) = 45.

Let me know if you'd like more details on any of these problems or have any questions!


Related Questions:

  1. How do you find the domain of composite functions?
  2. What is the significance of function composition in real-life applications?
  3. Can every function be composed with another function?
  4. How can we reverse a composition of functions?
  5. What happens when you compose two linear functions?

Tip: Always evaluate the inner function first when performing function compositions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Composition
Polynomial Functions

Formulas

(f ∘ g)(x) = f(g(x))

Theorems

Composition of Functions Theorem

Suitable Grade Level

Grade 10-12