Question

evaluating a rational function

what happens as x approaches zero from the positive direction?
f(x) gets larger

what happens as x approaches zero from the negative direction?
f(x) gets larger in the negative direction

what is the domain of the function?
all real numbers x
all positive real numbers x
all negative real numbers x
all real numbers x, except x = 0

Explanation:

Analyze the behavior of the function near zero

The tables and questions describe the behavior of a rational function \(f(x)\) as \(x\) approaches zero from both sides:

• As \(x \to 0^+\) (from the positive direction: \(1 \to 0.1 \to 0.01 \to 0.001\)), \(f(x)\) gets larger (approaches \(\infty\)).
• As \(x \to 0^-\) (from the negative direction: \(-1 \to -0.1 \to -0.01 \to -0.001\)), \(f(x)\) gets larger in the negative direction (approaches \(-\infty\)).

Determine the domain of the function

This behavior indicates a vertical asymptote at \(x = 0\). Since the function values grow without bound as \(x\) approaches \(0\), the function is undefined at \(x = 0\). The function is defined for all other real values of \(x\) on either side of zero. Therefore, the domain of the function is all real numbers \(x\), except \(x = 0\).

Answer:

• all real numbers x
• all positive real numbers x
• all negative real numbers x
• all real numbers x, except x = 0 (Correct answer)