Math / Algebra

QuestionFind the domain of the function. $f(x)=\sqrt{6 x-54}$
The domain is $\square$ - (Type your answer in interval notation.)

Studdy Solution

STEP 1

What is this asking? We need to find all the valid inputs ( $x$ values) for the function $f(x) = \sqrt{6x - 54}$ , meaning all the $x$ values that don't break the function! Watch out! Square roots of negative numbers are not real numbers, so we need to make sure whatever is inside that square root is zero or positive!

STEP 2

1. Ensure the expression inside the square root is non-negative.
2. Solve the inequality.
3. Express the solution in interval notation.

STEP 3

Alright, so the stuff inside the square root, which is $6x - 54$ , has to be greater than or equal to zero.
Why? Because we can't take the square root of a negative number and get a real answer!
So, we set up the inequality: $6x - 54 \ge 0$

STEP 4

Let's isolate the term with $x$ by adding 54 to both sides of the inequality: $6x - 54 + 54 \ge 0 + 54$ $6x \ge 54$

STEP 5

Now, we'll divide both sides by 6 to get $x$ all by itself.
Remember, since 6 is positive, the inequality sign stays the same: $\frac{6x}{6} \ge \frac{54}{6}$ $x \ge 9$

STEP 6

So, $x$ has to be greater than or equal to 9.
In interval notation, we write this as $[9, \infty)$ .
The square bracket means 9 is included, and the parenthesis around infinity means infinity is not a specific number we can include.

STEP 7

The domain of the function $f(x) = \sqrt{6x - 54}$ is $[9, \infty)$ .

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QuestionFind the domain of the function. $f(x)=\sqrt{6 x-54}$
The domain is $\square$ - (Type your answer in interval notation.)

Studdy Solution

STEP 1

STEP 2

1. Ensure the expression inside the square root is non-negative.
2. Solve the inequality.
3. Express the solution in interval notation.

STEP 3

Alright, so the stuff inside the square root, which is $6x - 54$ , has to be greater than or equal to zero.
Why? Because we can't take the square root of a negative number and get a real answer!
So, we set up the inequality: $6x - 54 \ge 0$

STEP 4

Let's isolate the term with $x$ by adding 54 to both sides of the inequality: $6x - 54 + 54 \ge 0 + 54$ $6x \ge 54$

STEP 5

Now, we'll divide both sides by 6 to get $x$ all by itself.
Remember, since 6 is positive, the inequality sign stays the same: $\frac{6x}{6} \ge \frac{54}{6}$ $x \ge 9$

STEP 6

So, $x$ has to be greater than or equal to 9.
In interval notation, we write this as $[9, \infty)$ .
The square bracket means 9 is included, and the parenthesis around infinity means infinity is not a specific number we can include.

STEP 7

The domain of the function $f(x) = \sqrt{6x - 54}$ is $[9, \infty)$ .

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QuestionFind the domain of the function. f(x)=6x−54f(x)=\sqrt{6 x-54}f(x)=6x−54​The domain is □\square□ - (Type your answer in interval notation.)

Studdy Solution

STEP 1

STEP 2

1. Ensure the expression inside the square root is non-negative.2. Solve the inequality.3. Express the solution in interval notation.

STEP 3

Alright, so the stuff inside the square root, which is 6x−546x - 546x−54, has to be greater than or equal to zero.Why? Because we can't take the square root of a negative number and get a *real* answer!So, we set up the inequality: 6x−54≥06x - 54 \ge 06x−54≥0

STEP 4

Let's *isolate* the term with xxx by adding **54** to both sides of the inequality: 6x−54+54≥0+546x - 54 + 54 \ge 0 + 546x−54+54≥0+54 6x≥546x \ge 546x≥54

STEP 5

Now, we'll *divide* both sides by **6** to get xxx all by itself.Remember, since 6 is positive, the inequality sign *stays the same*: 6x6≥546\frac{6x}{6} \ge \frac{54}{6}66x​≥654​ x≥9x \ge 9x≥9

STEP 6

So, xxx has to be greater than or equal to **9**.In interval notation, we write this as [9,∞)[9, \infty)[9,∞).The square bracket means **9** is *included*, and the parenthesis around infinity means infinity is *not* a specific number we can include.

STEP 7

The domain of the function f(x)=6x−54f(x) = \sqrt{6x - 54}f(x)=6x−54​ is [9,∞)[9, \infty)[9,∞).

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QuestionFind the domain of the function. $f(x)=\sqrt{6 x-54}$
The domain is $\square$ - (Type your answer in interval notation.)

1. Ensure the expression inside the square root is non-negative.
2. Solve the inequality.
3. Express the solution in interval notation.

Alright, so the stuff inside the square root, which is $6x - 54$ , has to be greater than or equal to zero.
Why? Because we can't take the square root of a negative number and get a real answer!
So, we set up the inequality: $6x - 54 \ge 0$

Let's isolate the term with $x$ by adding 54 to both sides of the inequality: $6x - 54 + 54 \ge 0 + 54$ $6x \ge 54$

Now, we'll divide both sides by 6 to get $x$ all by itself.
Remember, since 6 is positive, the inequality sign stays the same: $\frac{6x}{6} \ge \frac{54}{6}$ $x \ge 9$

So, $x$ has to be greater than or equal to 9.
In interval notation, we write this as $[9, \infty)$ .
The square bracket means 9 is included, and the parenthesis around infinity means infinity is not a specific number we can include.

The domain of the function $f(x) = \sqrt{6x - 54}$ is $[9, \infty)$ .