Express the domain of f(x) = (2x + 1)/(x^4 - 5x^2 + 4) in interval notation.

Step-by-step solution

1. 1

   Identify the only restriction. The numerator is a polynomial, defined everywhere; the function fails only where the denominator is zero. So the task reduces to solving $x^4 - 5x^2 + 4 = 0$.

2. 2

   Solve the biquadratic with a substitution. Only even powers appear, so put $u = x^2$:

   $u^2 - 5u + 4 = 0 \quad\Longrightarrow\quad (u-1)(u-4) = 0 \quad\Longrightarrow\quad u = 1 \text{ or } u = 4$

3. 3

   Return to x. Each positive $u$ gives two real values of $x$:

   $x^2 = 1 \Rightarrow x = \pm1, \qquad x^2 = 4 \Rightarrow x = \pm2$

   Four excluded points in total — the maximum a quartic denominator can produce.

4. 4

   Remove the four points and write the intervals. Ordered on the number line, the excluded values are $-2, -1, 1, 2$, which cut $\mathbb{R}$ into five pieces:

   $(-\infty,-2) \cup (-2,-1) \cup (-1,1) \cup (1,2) \cup (2,\infty)$

5. 5

   Check one point in each interval. At $x = -3$: denominator $81 - 45 + 4 = 40 \ne 0$. At $x = -1.5$: $5.0625 - 11.25 + 4 = -2.1875$. At $x = 0$: $4$. At $x = 1.5$: $-2.1875$. At $x = 3$: $40$. None vanishes, confirming the five intervals are entirely inside the domain.