Find the domain of the rational function.
[tex]g(x)=\frac{x+4}{x^2+64}[/tex]
A. [tex] \{x \mid x \neq-8, x \neq 8\} [/tex]
B. [tex] \{x \mid x \neq-8, x \neq 8, x=-4\} [/tex]
C. all real numbers
D. [tex] \{x \mid x \neq 0, x \neq-64\} [/tex]
Answer (1)
Find values of x where the denominator x 2 + 64 equals zero.
Solve x 2 + 64 = 0 for x , which gives x 2 = − 64 .
Determine that there are no real solutions for x since the square of a real number cannot be negative.
Conclude that the domain of the function is all real numbers, since no real x values make the denominator zero. all real numbers
Explanation
Understanding the Problem We are asked to find the domain of the rational function g ( x ) = x 2 + 64 x + 4 . The domain of a rational function consists of all real numbers except for those values of x that make the denominator equal to zero. So, we need to find the values of x for which the denominator x 2 + 64 is equal to zero.
Setting the Denominator to Zero To find the values of x that make the denominator zero, we set x 2 + 64 = 0 and solve for x . This gives us x 2 = − 64 .
Solving for x Since we are looking for real number solutions, we need to determine if there are any real numbers x such that x 2 = − 64 . The square of any real number is non-negative. Therefore, there are no real numbers x that satisfy the equation x 2 = − 64 .
Determining the Domain Since there are no real values of x that make the denominator equal to zero, the domain of the function g ( x ) = x 2 + 64 x + 4 is all real numbers.
Examples
Consider designing a suspension bridge. The function g ( x ) = x 2 + 64 x + 4 might model the stress distribution along the bridge, where x represents the distance from a central point. Knowing the domain (all real numbers in this case) ensures that the model is valid for all points along the bridge, preventing any undefined stress values that could lead to structural failure. This ensures the bridge's safety and stability across its entire span.
