Math Problem Statement

f(x) is x^2+3 not x^2-3

Solution

Thank you for pointing that out. Let's correct the calculations using the correct function f(x)=x2+3f(x) = x^2 + 3. I'll go through each operation again.

Given Functions:

  • f(x)=x2+3f(x) = x^2 + 3
  • g(x)=10x9g(x) = 10x - 9

1. (f+g)(x)(f + g)(x)

(f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x) Substitute the functions: (f+g)(x)=(x2+3)+(10x9)(f + g)(x) = (x^2 + 3) + (10x - 9) Simplify: (f+g)(x)=x2+10x6(f + g)(x) = x^2 + 10x - 6

  • Domain: Both f(x)f(x) and g(x)g(x) are polynomials, which are defined for all real numbers.
  • Domain in interval notation: (,)(-\infty, \infty)

2. (fg)(x)(f - g)(x)

(fg)(x)=f(x)g(x)(f - g)(x) = f(x) - g(x) Substitute the functions: (fg)(x)=(x2+3)(10x9)(f - g)(x) = (x^2 + 3) - (10x - 9) Simplify: (fg)(x)=x210x+12(f - g)(x) = x^2 - 10x + 12

  • Domain: Since both f(x)f(x) and g(x)g(x) are defined for all real numbers, the domain is all real numbers.
  • Domain in interval notation: (,)(-\infty, \infty)

3. (fg)(x)(f \cdot g)(x)

(fg)(x)=f(x)g(x)(f \cdot g)(x) = f(x) \cdot g(x) Substitute the functions: (fg)(x)=(x2+3)(10x9)(f \cdot g)(x) = (x^2 + 3) \cdot (10x - 9) Expand: (fg)(x)=10x39x2+30x27(f \cdot g)(x) = 10x^3 - 9x^2 + 30x - 27

  • Domain: This is a polynomial, so it is defined for all real numbers.
  • Domain in interval notation: (,)(-\infty, \infty)

4. (fg)(x)\left(\frac{f}{g}\right)(x)

(fg)(x)=f(x)g(x)\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} Substitute the functions: (fg)(x)=x2+310x9\left(\frac{f}{g}\right)(x) = \frac{x^2 + 3}{10x - 9}

  • Domain: The domain is all real numbers except where the denominator g(x)=10x9g(x) = 10x - 9 is zero.
  • Set 10x9=010x - 9 = 0 and solve for xx: 10x=910x = 9 x=910x = \frac{9}{10}
  • Therefore, the domain excludes x=910x = \frac{9}{10}.
  • Domain in interval notation: (,910)(910,)(-\infty, \frac{9}{10}) \cup (\frac{9}{10}, \infty)

Please let me know if you have any questions or need further assistance!


Here are five related questions you might find helpful:

  1. What is the difference in domain for a polynomial function compared to a rational function?
  2. How would the operations differ if one of the functions involved a square root?
  3. How would the domain change if f(x)=1x2+3f(x) = \frac{1}{x^2 + 3}?
  4. Can you think of a scenario where the domain of (fg)(x)(f \cdot g)(x) would not be all real numbers?
  5. How do the operations on functions affect the range compared to the original functions?

Tip: Always double-check the expressions of the functions you're working with to avoid any errors in calculations, especially when simplifying expressions.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Rational Functions
Domain

Formulas

Polynomial addition
Polynomial subtraction
Polynomial multiplication
Rational function domain calculation

Theorems

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Suitable Grade Level

High School