Domain of h(x) = (x^2 + 1) / (x^2 + 4)

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Flashcards

Domain of h(x) = (x^2+1)/(x^2+4)

The domain of h(x) = (x^2 + 1) / (x^2 + 4) is all real numbers because the denominator (x^2 + 4) is always positive and never equals zero for any real number x.

Study Notes

  • The question asks for the domain of the function h(x) = (x^2 + 1) / (x^2 + 4).

Domain of a Function

  • The domain of a function is the set of all possible input values (x-values) for which the function is defined.
  • In simpler terms, it's the set of x-values that you can plug into the function without causing any mathematical errors.

Restrictions on the Domain

  • Certain operations in mathematics can lead to undefined results:
    • Division by zero: A fraction is undefined if the denominator is zero.
    • Square root of a negative number: The square root of a negative number is not a real number.
    • Logarithm of a non-positive number: The logarithm of zero or a negative number is undefined.

Analyzing h(x) = (x2 + 1) / (x2 + 4)

  • h(x) is a rational function, meaning it's a fraction where the numerator and denominator are polynomials.
  • The numerator is x^2 + 1, which is defined for all real numbers. Any real number squared is non-negative, and adding 1 keeps it positive.
  • The denominator is x^2 + 4. The key question is whether this denominator can ever be equal to zero.

Checking for Division by Zero

  • We need to find out if there's any real value of x for which x^2 + 4 = 0.
  • x^2 + 4 = 0 implies x^2 = -4.
  • Since the square of any real number is non-negative, x^2 cannot be equal to -4 for any real number x.
  • Therefore, the denominator x^2 + 4 will never be zero for any real value of x.

Conclusion

  • Since the denominator x^2 + 4 is never zero for any real number x, there are no restrictions on the domain of h(x).
  • This means that h(x) is defined for all real numbers.
  • The domain of h(x) is the set of all real numbers.

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