Find the domain of the following function.
\[
f(x)=\sqrt[4]{-x^{2}+4 x+5}
\]
Choose the correct format for the domain (one of the four choices), then enter the corresponding values. Round to two decimal places where necessary.
$x< $
$x> $
\[
\leq x \leq
\]
$x \leq$
OR $\leq x$
Answer
Since we know that the terms inside any root need to be greater than or equal to zero, both $-x^2+4x+5\ge0$ must hold. This can be factored as $-(x-1)(x-5)\ge0$, the values of $x$ such that $-x^2+4x+5 \ge 0$ is $x \le 1$ or $x \ge 5$. Therefore, the domain of $f(x)$ is $x\in\boxed{(-\infty,1]\cup[5,\infty)}$
Step 1 ：Since we know that the terms inside any root need to be greater than or equal to zero, both $-x^2+4x+5\ge0$ must hold. This can be factored as $-(x-1)(x-5)\ge0$, the values of $x$ such that $-x^2+4x+5 \ge 0$ is $x \le 1$ or $x \ge 5$. Therefore, the domain of $f(x)$ is $x\in\boxed{(-\infty,1]\cup[5,\infty)}$
