Math / Algebra

QuestionFind the domain of the function $f(x) = \sqrt{1-x} + \sqrt{x-1}$ .

Studdy Solution

STEP 1

1. The function $f(x) = \sqrt{1-x} + \sqrt{x-1}$ involves two square roots.
2. The domain of a function involving square roots is determined by the non-negativity of the radicands.
3. We will find intervals where each radicand is non-negative.

STEP 2

1. Identify the radicands.
2. Determine the interval where $1 - x \geq 0$ .
3. Determine the interval where $x - 1 \geq 0$ .
4. Find the intersection of the intervals.

STEP 3

Identify the radicands of the function. The radicands are the expressions inside the square roots:
$(1-x) \quad \text{and} \quad (x-1)$

STEP 4

Determine the interval where $1 - x \geq 0$ :
$1 - x \geq 0$ $1 \geq x$
This means $x$ must be less than or equal to 1.

STEP 5

Determine the interval where $x - 1 \geq 0$ :
$x - 1 \geq 0$ $x \geq 1$
This means $x$ must be greater than or equal to 1.

STEP 6

Find the intersection of the intervals $1 \geq x$ and $x \geq 1$ . The intersection is where both conditions are satisfied simultaneously:
The intersection is:
$x = 1$
This means the only value of $x$ that satisfies both conditions is $x = 1$ .
The domain of the function is $\boxed{\{1\}}$ .

Was this helpful?

QuestionFind the domain of the function $f(x) = \sqrt{1-x} + \sqrt{x-1}$ .

Studdy Solution

STEP 1

1. The function $f(x) = \sqrt{1-x} + \sqrt{x-1}$ involves two square roots.
2. The domain of a function involving square roots is determined by the non-negativity of the radicands.
3. We will find intervals where each radicand is non-negative.

STEP 2

1. Identify the radicands.
2. Determine the interval where $1 - x \geq 0$ .
3. Determine the interval where $x - 1 \geq 0$ .
4. Find the intersection of the intervals.

STEP 3

Identify the radicands of the function. The radicands are the expressions inside the square roots:
$(1-x) \quad \text{and} \quad (x-1)$

STEP 4

Determine the interval where $1 - x \geq 0$ :
$1 - x \geq 0$ $1 \geq x$
This means $x$ must be less than or equal to 1.

STEP 5

Determine the interval where $x - 1 \geq 0$ :
$x - 1 \geq 0$ $x \geq 1$
This means $x$ must be greater than or equal to 1.

STEP 6

Find the intersection of the intervals $1 \geq x$ and $x \geq 1$ . The intersection is where both conditions are satisfied simultaneously:
The intersection is:
$x = 1$
This means the only value of $x$ that satisfies both conditions is $x = 1$ .
The domain of the function is $\boxed{\{1\}}$ .

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.

QuestionFind the domain of the function f(x)=1−x+x−1 f(x) = \sqrt{1-x} + \sqrt{x-1} f(x)=1−x​+x−1​.

Studdy Solution

STEP 1

1. The function f(x)=1−x+x−1 f(x) = \sqrt{1-x} + \sqrt{x-1} f(x)=1−x​+x−1​ involves two square roots.2. The domain of a function involving square roots is determined by the non-negativity of the radicands.3. We will find intervals where each radicand is non-negative.

STEP 2

1. Identify the radicands.2. Determine the interval where 1−x≥0 1 - x \geq 0 1−x≥0.3. Determine the interval where x−1≥0 x - 1 \geq 0 x−1≥0.4. Find the intersection of the intervals.

STEP 3

Identify the radicands of the function. The radicands are the expressions inside the square roots:(1−x)and(x−1) (1-x) \quad \text{and} \quad (x-1) (1−x)and(x−1)

STEP 4

Determine the interval where 1−x≥0 1 - x \geq 0 1−x≥0:1−x≥0 1 - x \geq 0 1−x≥0 1≥x 1 \geq x 1≥xThis means x x x must be less than or equal to 1.

STEP 5

Determine the interval where x−1≥0 x - 1 \geq 0 x−1≥0:x−1≥0 x - 1 \geq 0 x−1≥0 x≥1 x \geq 1 x≥1This means x x x must be greater than or equal to 1.

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.

QuestionFind the domain of the function $f(x) = \sqrt{1-x} + \sqrt{x-1}$ .

1. The function $f(x) = \sqrt{1-x} + \sqrt{x-1}$ involves two square roots.
2. The domain of a function involving square roots is determined by the non-negativity of the radicands.
3. We will find intervals where each radicand is non-negative.

1. Identify the radicands.
2. Determine the interval where $1 - x \geq 0$ .
3. Determine the interval where $x - 1 \geq 0$ .
4. Find the intersection of the intervals.

Identify the radicands of the function. The radicands are the expressions inside the square roots:
$(1-x) \quad \text{and} \quad (x-1)$

Determine the interval where $1 - x \geq 0$ :
$1 - x \geq 0$ $1 \geq x$
This means $x$ must be less than or equal to 1.

Determine the interval where $x - 1 \geq 0$ :
$x - 1 \geq 0$ $x \geq 1$
This means $x$ must be greater than or equal to 1.