Math Problem Statement

Suppose f(x)=x^2-x-12, g(x)=x+3, and h(x)=6x-1. Which of the following function operations are correct? Select three that apply. A: (f ÷ g)(x) = x - 4 B: (f - g)(x) = x^2 - 7x - 13 C: (g ⋅ h)(x) = 6x^2 + 17x - 3 D: (g - h)(x) = 5x - 4 E: (h + f)(x) = x^2 + 5x - 13

Solution

To determine which of the function operations are correct, let’s evaluate the provided functions step by step.

Given Functions:

  • f(x)=x2x12f(x) = x^2 - x - 12
  • g(x)=x+3g(x) = x + 3
  • h(x)=6x1h(x) = 6x - 1

Option A: (f÷g)(x)=x4(f \div g)(x) = x - 4

We divide f(x)f(x) by g(x)g(x):
f(x)g(x)=x2x12x+3\frac{f(x)}{g(x)} = \frac{x^2 - x - 12}{x + 3}
We perform polynomial long division:

  1. Divide x2x^2 by xx, which gives xx.
  2. Multiply back: x(x+3)=x2+3xx \cdot (x + 3) = x^2 + 3x.
  3. Subtract: (x2x12)(x2+3x)=4x12(x^2 - x - 12) - (x^2 + 3x) = -4x - 12.
  4. Divide 4x-4x by xx, which gives 4-4.
  5. Multiply back: 4(x+3)=4x12-4 \cdot (x + 3) = -4x - 12.
  6. Subtract: (4x12)(4x12)=0(-4x - 12) - (-4x - 12) = 0.

Thus,
f(x)g(x)=x4.\frac{f(x)}{g(x)} = x - 4.
Option A is correct.


Option B: (fg)(x)=x27x13(f - g)(x) = x^2 - 7x - 13

We subtract g(x)g(x) from f(x)f(x):
f(x)g(x)=(x2x12)(x+3).f(x) - g(x) = (x^2 - x - 12) - (x + 3).
Simplify:
f(x)g(x)=x2x12x3=x22x15.f(x) - g(x) = x^2 - x - 12 - x - 3 = x^2 - 2x - 15.
Thus, fgx27x13f - g \neq x^2 - 7x - 13.
Option B is incorrect.


Option C: (gh)(x)=6x2+17x3(g \cdot h)(x) = 6x^2 + 17x - 3

We multiply g(x)g(x) and h(x)h(x):
g(x)h(x)=(x+3)(6x1).g(x) \cdot h(x) = (x + 3)(6x - 1).
Use the distributive property:
g(x)h(x)=x(6x1)+3(6x1).g(x) \cdot h(x) = x(6x - 1) + 3(6x - 1).
Simplify:
g(x)h(x)=6x2x+18x3.g(x) \cdot h(x) = 6x^2 - x + 18x - 3.
Combine like terms:
g(x)h(x)=6x2+17x3.g(x) \cdot h(x) = 6x^2 + 17x - 3.
Option C is correct.


Option D: (gh)(x)=5x4(g - h)(x) = 5x - 4

We subtract h(x)h(x) from g(x)g(x):
g(x)h(x)=(x+3)(6x1).g(x) - h(x) = (x + 3) - (6x - 1).
Simplify:
g(x)h(x)=x+36x+1=5x+4.g(x) - h(x) = x + 3 - 6x + 1 = -5x + 4.
Thus, gh5x4g - h \neq 5x - 4.
Option D is incorrect.


Option E: (h+f)(x)=x2+5x13(h + f)(x) = x^2 + 5x - 13

We add h(x)h(x) to f(x)f(x):
h(x)+f(x)=(6x1)+(x2x12).h(x) + f(x) = (6x - 1) + (x^2 - x - 12).
Simplify:
h(x)+f(x)=x2x+6x112.h(x) + f(x) = x^2 - x + 6x - 1 - 12.
Combine like terms:
h(x)+f(x)=x2+5x13.h(x) + f(x) = x^2 + 5x - 13.
Option E is correct.


Final Answers:

The correct options are:
A, C, and E.


Do you need further clarifications or step-by-step solutions for any part? Let me know!

5 Related Questions:

  1. How do you perform polynomial long division step-by-step?
  2. How do you combine like terms when adding or subtracting polynomials?
  3. How do you distribute terms when multiplying binomials?
  4. What is the difference between function addition and subtraction?
  5. How can you verify the result of polynomial division?

Tip:

Always simplify expressions and combine like terms carefully to avoid errors in polynomial operations.

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Math Problem Analysis

Mathematical Concepts

Polynomial Operations
Function Addition
Function Subtraction
Polynomial Multiplication
Polynomial Division

Formulas

Polynomial Division: (f ÷ g)(x)
Polynomial Addition/Subtraction: f(x) ± g(x)
Polynomial Multiplication: g(x) ⋅ h(x)

Theorems

Distributive Property
Polynomial Division Theorem

Suitable Grade Level

Grades 9-11