Find the domain of each rational function. f(x)=(x+7)/(x²+49)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{x+7}{x^{2} + 49}\).
2. Recall that the domain of a rational function includes all real numbers except where the denominator is zero, because division by zero is undefined.
3. Set the denominator equal to zero to find values to exclude: \(x^{2} + 49 = 0\).
4. Solve the equation \(x^{2} + 49 = 0\) by isolating \(x^{2}\): \(x^{2} = -49\).
5. Since \(x^{2} = -49\) has no real solutions (because a square cannot be negative in the real number system), the denominator is never zero for any real \(x\), so the domain is all real numbers.

Key Concepts

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. For rational functions, the domain excludes any x-values that make the denominator zero, as division by zero is undefined.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of the numerator and denominator polynomials is essential, especially identifying values that cause the denominator to be zero.

Solving Quadratic Equations

To find values that make the denominator zero, you solve the quadratic equation Q(x) = 0. Techniques include factoring, completing the square, or using the quadratic formula to determine if real roots exist that restrict the domain.