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

Verified step by step guidance

1. Identify the rational function given: \(h(x) = \frac{x+7}{x^{2} - 49}\).
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 the values to exclude: \(x^{2} - 49 = 0\).
4. Solve the equation \(x^{2} - 49 = 0\) by factoring it as a difference of squares: \((x - 7)(x + 7) = 0\).
5. Find the roots from the factors: \(x - 7 = 0\) gives \(x = 7\), and \(x + 7 = 0\) gives \(x = -7\). These values are excluded from the domain.

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational functions, the domain excludes 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 how to simplify and analyze these functions is essential, especially identifying values that cause the denominator Q(x) to be zero.

Factoring Polynomials

Factoring involves rewriting polynomials as products of simpler polynomials. For example, x² - 49 factors as (x - 7)(x + 7). Factoring helps identify zeros of the denominator to determine excluded values from the domain.