f$($x$)$

$=$

$$

$3$

$+$

$3$

$2$

$$

$2$

$$

$1$

$8$

$$

$+$

$4$

$.$

$=$x

$3$

$+$

$2$

$3$

$$

x

$2$

$18$x$+4.$

Over which of these domains could you define f $-1($x$)?$

Remember that f$($x$)$ must be one$-$to$-$one $($only one y value for each x value$)$ over the domain where f $-1($x$)$ is defined as a function. So$,$ in some cases you must restrict the domain of f$($x$)$ so that it's one$-$to$-$one. There might be more than one section of domain that's one$-$to$-$one.

HINT: The function f$($x$)$ is one$-$to$-$one wherever it's monotonic $("$monotonic$"$ simply means increasing or decreasing over the entire interval, so the derivative does not change sign$).$

To find these intervals, start by taking the first derivative, which is

$$

$$

$($

$$

$)$

$=$

$3$

$$

$2$

$+$

$3$

$$

$$

$1$

$8$

$,$

f

$$

$($x$)=3$x

$2$

$+3$x$18,$ and setting it equal to zero.

$($

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$,$

$$

$3$

$]$

$($$,3]$

$[$

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$3$

$,$

$)$

$[3,$$)$

$[$

$2$

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$)$

$[2,$$)$