Math Problem Statement

Evaluate each expression using the values given in the table.

Solution

Let's evaluate the expressions using the table provided.

Given:

  • x values: 3,2,1,0,1,2,3-3, -2, -1, 0, 1, 2, 3
  • f(x) values: 10,8,6,4,2,0,210, 8, 6, 4, 2, 0, -2
  • g(x) values: 9,3,0,1,0,3,9-9, -3, 0, 1, 0, -3, -9

The questions involve compositions of functions, which follow the rule:
(fg)(x)=f(g(x))and(gf)(x)=g(f(x))(f \circ g)(x) = f(g(x))\quad \text{and}\quad (g \circ f)(x) = g(f(x))

Now, let's solve the expressions step-by-step.


a. (fg)(1)(f \circ g)(1)

  1. Find g(1)g(1): From the table, g(1)=0g(1) = 0.
  2. Now, find f(0)f(0): From the table, f(0)=4f(0) = 4.
    • So, (fg)(1)=f(g(1))=f(0)=4(f \circ g)(1) = f(g(1)) = f(0) = 4.

b. (fg)(2)(f \circ g)(2)

  1. Find g(2)g(2): From the table, g(2)=3g(2) = -3.
  2. Now, find f(3)f(-3): From the table, f(3)=10f(-3) = 10.
    • So, (fg)(2)=f(g(2))=f(3)=10(f \circ g)(2) = f(g(2)) = f(-3) = 10.

c. (gf)(2)(g \circ f)(2)

  1. Find f(2)f(2): From the table, f(2)=0f(2) = 0.
  2. Now, find g(0)g(0): From the table, g(0)=1g(0) = 1.
    • So, (gf)(2)=g(f(2))=g(0)=1(g \circ f)(2) = g(f(2)) = g(0) = 1.

d. (gf)(3)(g \circ f)(3)

  1. Find f(3)f(3): From the table, f(3)=2f(3) = -2.
  2. Now, find g(2)g(-2): From the table, g(2)=3g(-2) = -3.
    • So, (gf)(3)=g(f(3))=g(2)=3(g \circ f)(3) = g(f(3)) = g(-2) = -3.

e. (gg)(1)(g \circ g)(1)

  1. Find g(1)g(1): From the table, g(1)=0g(1) = 0.
  2. Now, find g(0)g(0): From the table, g(0)=1g(0) = 1.
    • So, (gg)(1)=g(g(1))=g(0)=1(g \circ g)(1) = g(g(1)) = g(0) = 1.

f. (ff)(3)(f \circ f)(3)

  1. Find f(3)f(3): From the table, f(3)=2f(3) = -2.
  2. Now, find f(2)f(-2): From the table, f(2)=8f(-2) = 8.
    • So, (ff)(3)=f(f(3))=f(2)=8(f \circ f)(3) = f(f(3)) = f(-2) = 8.

Final Answers:

a. (fg)(1)=4(f \circ g)(1) = 4
b. (fg)(2)=10(f \circ g)(2) = 10
c. (gf)(2)=1(g \circ f)(2) = 1
d. (gf)(3)=3(g \circ f)(3) = -3
e. (gg)(1)=1(g \circ g)(1) = 1
f. (ff)(3)=8(f \circ f)(3) = 8


Would you like to explore any of these steps further?


5 Related Questions:

  1. How do you identify the domain and range of a function?
  2. What is the difference between f(g(x))f(g(x)) and g(f(x))g(f(x))?
  3. How do you compose more than two functions, like h(f(g(x)))h(f(g(x)))?
  4. What is the significance of function composition in real-world applications?
  5. How can we verify if a function is invertible through composition?

Tip:
When working with function compositions, always solve the inner function first before moving to the outer function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra

Formulas

(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))

Theorems

Function Composition

Suitable Grade Level

Grades 9-11