Determine if the following relations define $$ y $$ as a function of $$ x $$. (a) $$ x=|5 y| $$ (b) $$ y=|5 x| $$ Part 1 of 2 $$ x=|5 y| $$ (Choose one) $$
abla $$ a function. Part 2 of 2 $$ y=|5 x| $$ (Choose one) $$
abla $$ a function.

Question

Determine if the following relations define $$ y $$ as a function of $$ x $$. (a) $$ x=|5 y| $$ (b) $$ y=|5 x| $$ Part 1 of 2 $$ x=|5 y| $$ (Choose one) $$ 
abla $$ a function. Part 2 of 2 $$ y=|5 x| $$ (Choose one) $$ 
abla $$ a function.

UpStudy AI · Accepted Answer

Let's analyze each relation to determine whether $$ y $$ is a function of $$ x $$.

### Part 1: $$ x = |5y| $$

To determine if $$ y $$ is a function of $$ x $$, we need to see if each $$ x $$ corresponds to exactly one $$ y $$.

1. **Solve for $$ y $$:**
   $$
   x = |5y| \implies |y| = \frac{x}{5} \implies y = \pm\frac{x}{5}
   $$
   
2. **Interpretation:**
   - For each positive $$ x $$, there are two possible values of $$ y $$: $$ y = \frac{x}{5} $$ and $$ y = -\frac{x}{5} $$.
   - For $$ x = 0 $$, there is only one value: $$ y = 0 $$.

3. **Conclusion:**
   Since a single $$ x $$ (for $$ x > 0 $$) corresponds to two different $$ y $$ values, $$ y $$ is **not** uniquely determined by $$ x $$. Therefore, this relation does **not** define $$ y $$ as a function of $$ x $$.

**Answer for Part 1:**  
$$ x = |5y| $$ does **not** define $$ y $$ as a function of $$ x $$.

---

### Part 2: $$ y = |5x| $$

Now, let's determine if $$ y $$ is a function of $$ x $$.

1. **Solve for $$ y $$:**
   $$
   y = |5x|
   $$
   
2. **Interpretation:**
   - For every real number $$ x $$, $$ y $$ is defined as the absolute value of $$ 5x $$.
   - This means for each $$ x $$, there is exactly one corresponding $$ y $$.

3. **Conclusion:**
   Since each $$ x $$ maps to exactly one $$ y $$, $$ y $$ is uniquely determined by $$ x $$. Therefore, this relation **does** define $$ y $$ as a function of $$ x $$.

**Answer for Part 2:**  
$$ y = |5x| $$ **defines** $$ y $$ as a function of $$ x $$.

---

**Summary:**
- **Part 1:** $$ x = |5y| $$ is **not** a function.
- **Part 2:** $$ y = |5x| $$ **is** a function.

**Final Answer:**

- **Part 1:** $$ x = |5y| $$ does **not** define $$ y $$ as a function.
- **Part 2:** $$ y = |5x| $$ **does** define $$ y $$ as a function.
- **Part 1:** $$ x = |5y| $$ does **not** define $$ y $$ as a function. - **Part 2:** $$ y = |5x| $$ **does** define $$ y $$ as a function.

Determine if the following relations define \( y \) as a function of \( x \). (a) \( x=|5 y| \) (b) \( y=|5 x| \) Part 1 of 2 \( x=|5 y| \) (Choose one) \( \nabla \) a function. Part 2 of 2 \( y=|5 x| \) (Choose one) \( \nabla \) a function.

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