Question

1. graph the function and identify all key characteristics: $f(x) = (x - 1)(x + 1)^3(x$domain: $(-\infty,\infty)$range: $-4,\infty$end behavior: as. $x \to \infty, f(x) \to +\infty$$x \to -\infty, f(x) \to +\infty$inc. intervals: $0,1.5 2,\infty)$dec. intervals: _______________zeros: $x=1$ $x=-1$ $x=2$| $x$ | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 ||-----|----|----|---|---|---|---|---|---|| $p(x)$ | 48 | 0 | -4 | 0 | 0 | 128 | 0 | |

Explanation:

Step1: Identify missing factor of f(x)

From zeros $x=1, x=-1, x=2$, the full function is $f(x)=(x-1)(x+1)^3(x-2)$.

Step2: Analyze decreasing intervals via graph

The graph falls from $(-\infty,0)$, then from $(1.5,2)$.

Step3: Confirm with function values

At $x=-2$, $f(x)=48$; $x=0$, $f(x)=-4$ (decreasing). At $x=1.5$, $f(x)$ rises to a peak, then falls to $x=2$ where $f(x)=0$.

Answer:

Dec. Intervals: $(-\infty, 0), (1.5, 2)$