In Exercises 1–8, find the domain of each rational function. h(x)=(x+7)/(x²−49)

Verified step by step guidance

1. Identify the rational function given: $h\left(x\right)=\frac{x+7}{x^2-49}$.
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 as a difference of squares: $(x\; -\; 7)(x\; +\; 7)\; =\; 0$. Then find the roots by setting each factor equal to zero.
5. Exclude the values found from the domain. The domain is all real numbers except $x\; =\; 7$ and $x\; =\; -7$.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding rational functions involves recognizing that the function is undefined where the denominator equals zero, which affects the domain.

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.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This is essential for identifying values that make the denominator zero, such as factoring x^2 - 49 into (x - 7)(x + 7) to find restrictions on the domain.