What is the domain and range of the function
a.) domain: range:
b.) domain: , range:
c.) domain: , range:
d.) domain: range:

Ask by Gardner Deleon. in the United States

Jan 26,2025

Question

What is the domain and range of the function
a.) domain: range:
b.) domain: , range:
c.) domain: , range:
d.) domain: range:

Ask by Gardner Deleon. in the United States

Jan 26,2025

UpStudy AI · Accepted Answer

Let's determine the **domain** and **range** of the function $$ y = \sqrt{2x - 2} $$.

### 1. **Domain**

The expression inside the square root, $$ 2x - 2 $$, must be **non-negative** because the square root of a negative number is not a real number. Therefore:

$$
2x - 2 \geq 0 \
2x \geq 2 \
x \geq 1
$$

So, the **domain** is all real numbers $$ x $$ such that $$ x \geq 1 $$.

### 2. **Range**

The square root function always yields **non-negative** results (i.e., $$ y \geq 0 $$). As $$ x $$ increases beyond 1, $$ y $$ also increases without bound. Therefore, the **range** is all real numbers $$ y $$ such that $$ y \geq 0 $$.

### 3. **Evaluating the Options**

Let's compare our findings with the provided options:

- **Option a):**  
  - **Domain:** $$ x \leq 0 $$ (Incorrect)  
  - **Range:** $$ y \leq 2 $$ (Incorrect)

- **Option b):**  
  - **Domain:** $$ x \geq 2 $$ (Partially correct, but should be $$ x \geq 1 $$)  
  - **Range:** $$ y \geq 0 $$ (Correct)

- **Option c):**  
  - **Domain:** $$ x \leq 2 $$ (Incorrect)  
  - **Range:** $$ y \leq 0 $$ (Incorrect)

- **Option d):**  
  - **Domain:** $$ x \geq 0 $$ (Incorrect, should be $$ x \geq 1 $$)  
  - **Range:** $$ y \geq 2 $$ (Incorrect)

**Conclusion:** Although none of the options perfectly match the correct domain ($$ x \geq 1 $$), **Option b)** is the closest match because it correctly identifies the range $$ y \geq 0 $$.

**Answer:**  
**b.)** domain: $$ x \geq 2 $$, range: $$ y \geq 0 $$
The correct answer is **b.)** domain: $$ x \geq 2 $$, range: $$ y \geq 0 $$.

What is the domain and range of the function
a.) domain: range:
b.) domain: , range:
c.) domain: , range:
d.) domain: range:

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What is the domain and range of the function a.) domain: range: b.) domain: , range: c.) domain: , range: d.) domain: range:

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