QuestionFind the domain of the function $f(x)=\sqrt[4]{17-x}$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\sqrt[4]{17-x}$ . The domain of a function is the set of all possible input values (x-values) which will produce a valid output.
3. For a square root (or any even root), the radicand (the value under the root) must be greater than or equal to zero, because you can't take the square root of a negative number and get a real number.

STEP 2

We need to find the values of $x$ for which the expression under the root, $17-x$ , is greater than or equal to zero. This is because the fourth root of a negative number is not defined in the real number system.
$17-x \geq0$

STEP 3

To solve this inequality, we need to isolate $x$ . We can do this by subtracting $17$ from both sides of the equation.
$-x \geq -17$

STEP 4

Now, we multiply both sides by $-1$ to get $x$ alone on one side. Remember, when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality.
$x \leq17$

STEP 5

So, the domain of the function $f(x)=\sqrt[4]{17-x}$ is all real numbers $x$ such that $x \leq17$ . In interval notation, this is written as $(-\infty,17]$ .
The domain of the function is $(-\infty,17]$ .

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