Math / Algebra

QuestionFind the domain of $f$ . $f(x)=\frac{x-4}{x^{2}+6 x}$ A. $\{x \mid x \neq 6\}$ B. $\{x \mid x \neq 0, x \neq-6\}$ C. $\{x \mid x \neq 4\}$ D. $\{x \mid x \neq-6\}$

Studdy Solution

STEP 1

What is this asking? We need to find all the allowable $x$ values for the function $f(x)$ . Watch out! Remember, we can't divide by zero!

STEP 2

1. Find the values that make the denominator zero.
2. Exclude those values from the domain.

STEP 3

Alright, so we've got this function, and the only thing we need to watch out for is when the denominator is zero.
So, let's set that denominator, $x^2 + 6x$ , equal to zero and solve! $x^2 + 6x = 0$

STEP 4

We can factor out an $x$ from both terms: $x(x + 6) = 0$

STEP 5

Now, we have two factors multiplying to zero.
This means either $x$ is zero or $x + 6$ is zero.
If $x$ is zero, well, $x$ is zero!
If $x + 6$ is zero, we subtract 6 from both sides to get $x = \mathbf{-6}$ .

STEP 6

So, we found that $x$ cannot be zero, and $x$ cannot be $-6$ .
Any other value of $x$ is perfectly fine!
So, the domain of $f(x)$ is all $x$ except for $\mathbf{0}$ and $\mathbf{-6}$ .

STEP 7

The domain of $f$ is $\{x \mid x \neq 0, x \neq -6\}$ , which is option B.

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QuestionFind the domain of $f$ . $f(x)=\frac{x-4}{x^{2}+6 x}$ A. $\{x \mid x \neq 6\}$ B. $\{x \mid x \neq 0, x \neq-6\}$ C. $\{x \mid x \neq 4\}$ D. $\{x \mid x \neq-6\}$

Studdy Solution

STEP 1

What is this asking? We need to find all the allowable $x$ values for the function $f(x)$ . Watch out! Remember, we can't divide by zero!

STEP 2

1. Find the values that make the denominator zero.
2. Exclude those values from the domain.

STEP 3

Alright, so we've got this function, and the only thing we need to watch out for is when the denominator is zero.
So, let's set that denominator, $x^2 + 6x$ , equal to zero and solve! $x^2 + 6x = 0$

STEP 4

We can factor out an $x$ from both terms: $x(x + 6) = 0$

STEP 5

Now, we have two factors multiplying to zero.
This means either $x$ is zero or $x + 6$ is zero.
If $x$ is zero, well, $x$ is zero!
If $x + 6$ is zero, we subtract 6 from both sides to get $x = \mathbf{-6}$ .

STEP 6

So, we found that $x$ cannot be zero, and $x$ cannot be $-6$ .
Any other value of $x$ is perfectly fine!
So, the domain of $f(x)$ is all $x$ except for $\mathbf{0}$ and $\mathbf{-6}$ .

STEP 7

The domain of $f$ is $\{x \mid x \neq 0, x \neq -6\}$ , which is option B.

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Studdy Solution

STEP 1

What is this asking? We need to find all the *allowable* xxx values for the function f(x)f(x)f(x). Watch out! Remember, we can't divide by **zero**!

STEP 2

1. Find the values that make the denominator zero.2. Exclude those values from the domain.

STEP 3

Alright, so we've got this function, and the *only* thing we need to watch out for is when the denominator is zero.So, let's **set** that denominator, x2+6xx^2 + 6xx2+6x, equal to **zero** and *solve*! x2+6x=0x^2 + 6x = 0x2+6x=0

STEP 4

We can **factor out** an xxx from both terms: x(x+6)=0x(x + 6) = 0x(x+6)=0

STEP 5

Now, we have two factors multiplying to zero.This means *either* xxx is zero *or* x+6x + 6x+6 is zero.If xxx is zero, well, xxx is **zero**!If x+6x + 6x+6 is zero, we subtract 6 from both sides to get x=−6x = \mathbf{-6}x=−6.

STEP 6

So, we found that xxx *cannot* be zero, and xxx *cannot* be −6-6−6.Any other value of xxx is perfectly fine!So, the **domain** of f(x)f(x)f(x) is all xxx *except* for 0\mathbf{0}0 and −6\mathbf{-6}−6.

STEP 7

The domain of fff is {x∣x≠0,x≠−6}\{x \mid x \neq 0, x \neq -6\}{x∣x=0,x=−6}, which is option **B**.

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What is this asking? We need to find all the allowable $x$ values for the function $f(x)$ . Watch out! Remember, we can't divide by zero!

1. Find the values that make the denominator zero.
2. Exclude those values from the domain.

Alright, so we've got this function, and the only thing we need to watch out for is when the denominator is zero.
So, let's set that denominator, $x^2 + 6x$ , equal to zero and solve! $x^2 + 6x = 0$

We can factor out an $x$ from both terms: $x(x + 6) = 0$

Now, we have two factors multiplying to zero.
This means either $x$ is zero or $x + 6$ is zero.
If $x$ is zero, well, $x$ is zero!
If $x + 6$ is zero, we subtract 6 from both sides to get $x = \mathbf{-6}$ .

So, we found that $x$ cannot be zero, and $x$ cannot be $-6$ .
Any other value of $x$ is perfectly fine!
So, the domain of $f(x)$ is all $x$ except for $\mathbf{0}$ and $\mathbf{-6}$ .

The domain of $f$ is $\{x \mid x \neq 0, x \neq -6\}$ , which is option B.