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

Verified step by step guidance

1. Identify the rational function given: $f\left(x\right)=\frac{x+7}{x^2+49}$.
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 values to exclude: $x^2+49=0$.
4. Solve the equation $x^2+49=0$ for $x$ to find any real roots that would make the denominator zero.
5. Since the denominator has no real roots (because $x^2+\; 49\; =\; 0$ has no real solutions), conclude that the domain is all real numbers.

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 Q(x) 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, since division by zero is undefined.

Solving Quadratic Equations

To find where the denominator is zero, you solve the quadratic equation Q(x) = 0. This involves factoring, completing the square, or using the quadratic formula to identify values that must be excluded from the domain.