Math / Algebra

Question5.
Describe the domain for the following function. Write your answer in interval notation. $f(x)=\frac{1}{x-7}-\frac{\sqrt{x+2}}{x-6}$

Studdy Solution

STEP 1

1. The function $f(x) = \frac{1}{x-7} - \frac{\sqrt{x+2}}{x-6}$ involves a rational expression and a square root.
2. The domain of the function is restricted by the values that make the denominators zero and the radicand negative.
3. We will need to find intervals where the denominators are non-zero and the radicand is non-negative.

STEP 2

1. Identify the restrictions from the denominators.
2. Identify the restrictions from the square root.
3. Determine the domain by combining the restrictions.
4. Write the domain in interval notation.

STEP 3

Identify the restrictions from the denominators:
1. The denominator $x - 7$ cannot be zero, so: $x \neq 7$
2. The denominator $x - 6$ cannot be zero, so: $x \neq 6$

STEP 4

Identify the restrictions from the square root:
The expression under the square root $x + 2$ must be non-negative: $x + 2 \geq 0$ $x \geq -2$

STEP 5

Determine the domain by combining the restrictions:
1. From the square root, $x \geq -2$ .
2. From the denominators, $x \neq 6$ and $x \neq 7$ .
Combine these to find the domain: $x \geq -2 \quad \text{and} \quad x \neq 6, 7$

STEP 6

Write the domain in interval notation:
The domain is all $x$ such that $x$ is greater than or equal to $-2$ , excluding $6$ and $7$ . In interval notation, this is:
$[-2, 6) \cup (6, 7) \cup (7, \infty)$
The domain of the function is $\boxed{[-2, 6) \cup (6, 7) \cup (7, \infty)}$ .

Was this helpful?

Question5.
Describe the domain for the following function. Write your answer in interval notation. $f(x)=\frac{1}{x-7}-\frac{\sqrt{x+2}}{x-6}$

Studdy Solution

STEP 1

STEP 2

1. Identify the restrictions from the denominators.
2. Identify the restrictions from the square root.
3. Determine the domain by combining the restrictions.
4. Write the domain in interval notation.

STEP 3

Identify the restrictions from the denominators:
1. The denominator $x - 7$ cannot be zero, so: $x \neq 7$
2. The denominator $x - 6$ cannot be zero, so: $x \neq 6$

STEP 4

Identify the restrictions from the square root:
The expression under the square root $x + 2$ must be non-negative: $x + 2 \geq 0$ $x \geq -2$

STEP 5

Determine the domain by combining the restrictions:
1. From the square root, $x \geq -2$ .
2. From the denominators, $x \neq 6$ and $x \neq 7$ .
Combine these to find the domain: $x \geq -2 \quad \text{and} \quad x \neq 6, 7$

STEP 6

Write the domain in interval notation:
The domain is all $x$ such that $x$ is greater than or equal to $-2$ , excluding $6$ and $7$ . In interval notation, this is:
$[-2, 6) \cup (6, 7) \cup (7, \infty)$
The domain of the function is $\boxed{[-2, 6) \cup (6, 7) \cup (7, \infty)}$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question5.Describe the domain for the following function. Write your answer in interval notation. f(x)=1x−7−x+2x−6f(x)=\frac{1}{x-7}-\frac{\sqrt{x+2}}{x-6}f(x)=x−71​−x−6x+2​​

Studdy Solution

STEP 1

STEP 2

1. Identify the restrictions from the denominators.2. Identify the restrictions from the square root.3. Determine the domain by combining the restrictions.4. Write the domain in interval notation.

STEP 3

Identify the restrictions from the denominators:1. The denominator x−7 x - 7 x−7 cannot be zero, so: x≠7 x \neq 7 x=72. The denominator x−6 x - 6 x−6 cannot be zero, so: x≠6 x \neq 6 x=6

STEP 4

Identify the restrictions from the square root:The expression under the square root x+2 x + 2 x+2 must be non-negative: x+2≥0 x + 2 \geq 0 x+2≥0 x≥−2 x \geq -2 x≥−2

STEP 5

Determine the domain by combining the restrictions:1. From the square root, x≥−2 x \geq -2 x≥−2.2. From the denominators, x≠6 x \neq 6 x=6 and x≠7 x \neq 7 x=7.Combine these to find the domain: x≥−2andx≠6,7 x \geq -2 \quad \text{and} \quad x \neq 6, 7 x≥−2andx=6,7

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question5.
Describe the domain for the following function. Write your answer in interval notation. $f(x)=\frac{1}{x-7}-\frac{\sqrt{x+2}}{x-6}$

1. Identify the restrictions from the denominators.
2. Identify the restrictions from the square root.
3. Determine the domain by combining the restrictions.
4. Write the domain in interval notation.

Identify the restrictions from the denominators:
1. The denominator $x - 7$ cannot be zero, so: $x \neq 7$
2. The denominator $x - 6$ cannot be zero, so: $x \neq 6$

Identify the restrictions from the square root:
The expression under the square root $x + 2$ must be non-negative: $x + 2 \geq 0$ $x \geq -2$

Determine the domain by combining the restrictions:
1. From the square root, $x \geq -2$ .
2. From the denominators, $x \neq 6$ and $x \neq 7$ .
Combine these to find the domain: $x \geq -2 \quad \text{and} \quad x \neq 6, 7$