Find the value of p and the equation of the horizontal asymptote for the function f(x) = a/(x+p) + q, given its domain is (−∞, 1) ∪ (1, ∞) and its range is (−∞, 2) ∪ (2, ∞).

May 14, 2026Mathematics

ScanSolve AI Answer

May 14, 2026

Answer

p = -1

Step 1: Determine the value of $p$ . The domain of the function $f(x) = \frac{a}{x+p} + q$ is given as $(-\infty; 1) \cup (1; \infty)$ . This indicates that there is a vertical asymptote at $x=1$ . For a rational function of this form, the vertical asymptote occurs when the denominator is zero, i.e., $x+p = 0$ . Therefore, $x = -p$ . Comparing this with $x=1$ , we have $-p = 1$ , which means $p = -1$ . $\boxed{p = -1}$

Step 2: Determine the equation of the horizontal asymptote of $f$ . The general form of

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Determine the value of p. The domain of the function f(x) = (a)/(x+p) + q is given as (-; 1) (1; ).

Asked on May 14, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on May 14, 2026

Step 1: Determine the value of p. The domain of the function f(x) = (a)/(x+p) + q is given as (-; 1) (1; ). This indicates that there is a vertical asymptote at x=1. For a rational function of this form, the vertical asymptote occurs when the denominator is zero, i.e., x+p = 0. Therefore, x = -p. Comparing this with x=1, we have -p = 1, which means p = -1. p = -1 Step 2: Determine the equation of the horizontal asymptote of f. The general form of