Find the domain of the rational function.
[tex]g(x)=\frac{x+5}{x^2-64}[/tex]
Answer (2)
Identify that the domain of a rational function excludes values where the denominator is zero.
Solve the equation x 2 − 64 = 0 by factoring it as ( x − 8 ) ( x + 8 ) = 0 .
Find the roots x = 8 and x = − 8 , which make the denominator zero.
Express the domain as all real numbers except 8 and − 8 : { x ∣ x = − 8 , x = 8 } .
Explanation
Understanding the Problem We are asked to find the domain of the rational function g ( x ) = x 2 − 64 x + 5 . The domain of a rational function consists of all real numbers except for the values of x that make the denominator equal to zero.
Finding the Zeros of the Denominator To find the values of x that make the denominator zero, we need to solve the equation x 2 − 64 = 0 .
Solving for x We can factor the quadratic expression as a difference of squares: x 2 − 64 = ( x − 8 ) ( x + 8 ) Setting each factor equal to zero gives us: x − 8 = 0 ⇒ x = 8 x + 8 = 0 ⇒ x = − 8
Identifying Values to Exclude Thus, the denominator is zero when x = 8 or x = − 8 . These are the values that must be excluded from the domain.
Expressing the Domain The domain of g ( x ) is all real numbers except x = 8 and x = − 8 . In set notation, this is written as: { x ∣ x = − 8 , x = 8 }
Examples
Rational functions are used in various fields, such as physics, engineering, and economics. For example, in physics, they can describe the relationship between voltage and current in an electrical circuit. In economics, they can model cost-benefit ratios or supply-demand curves. Understanding the domain of a rational function ensures that the model is valid and doesn't produce undefined results, such as dividing by zero, which is not physically or economically meaningful.
The domain of the function g ( x ) = x 2 − 64 x + 5 excludes the values that make the denominator zero. Specifically, the values to exclude are -8 and 8. Therefore, the domain can be expressed as { x ∣ x = − 8 , x = 8 } .
;
