Math Problem Statement

when f(x)=3x²-2 and g(x)=1/5x²=4 what is the domain of f+g

Solution

To determine the domain of f(x)+g(x)f(x) + g(x), we first need to understand the domains of f(x)f(x) and g(x)g(x) individually.

Given:

  • f(x)=3x22f(x) = 3x^2 - 2
  • g(x)=15x2+4g(x) = \frac{1}{5}x^2 + 4

Domain of f(x)f(x):

The function f(x)=3x22f(x) = 3x^2 - 2 is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of f(x)f(x) is all real numbers: Domain of f(x)=(,)\text{Domain of } f(x) = (-\infty, \infty)

Domain of g(x)g(x):

The function g(x)=15x2+4g(x) = \frac{1}{5}x^2 + 4 is also a polynomial (since it is a quadratic function). Polynomials are defined for all real numbers as well. So, the domain of g(x)g(x) is: Domain of g(x)=(,)\text{Domain of } g(x) = (-\infty, \infty)

Domain of f(x)+g(x)f(x) + g(x):

Since both f(x)f(x) and g(x)g(x) are defined for all real numbers, the domain of their sum f(x)+g(x)f(x) + g(x) is the intersection of their domains. Since both domains are (,)(-\infty, \infty), the domain of f(x)+g(x)f(x) + g(x) is also: Domain of f(x)+g(x)=(,)\text{Domain of } f(x) + g(x) = (-\infty, \infty)

This means that f(x)+g(x)f(x) + g(x) is defined for all real numbers.

Would you like further details or have any questions?

Here are some related questions you might find interesting:

  1. How do you determine the domain of a rational function?
  2. What is the range of f(x)+g(x)f(x) + g(x) given the functions above?
  3. How would you find the domain of a composite function like f(g(x))f(g(x))?
  4. What are the general rules for finding the domain of a function involving square roots?
  5. How do you find the domain of a function that includes both polynomials and square roots?

Tip: The domain of a function is the set of all possible input values (x-values) that will not cause any undefined behavior in the function, such as division by zero or taking the square root of a negative number.