Decide whether the given relation defines $$ y $$ as a function of $$ x $$. Give the domain and range. $$ y=\sqrt{4 x+1} $$ Does the relation define a function? Yes No

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To determine whether the relation $$ y = \sqrt{4x + 1} $$ defines $$ y $$ as a function of $$ x $$, we need to check if for every value of $$ x $$ in the domain, there is exactly one corresponding value of $$ y $$.

1. **Check if it defines a function:**
   - The expression $$ \sqrt{4x + 1} $$ is defined for values of $$ x $$ such that $$ 4x + 1 \geq 0 $$.
   - Solving the inequality:
     $$
     4x + 1 \geq 0 \implies 4x \geq -1 \implies x \geq -\frac{1}{4}
     $$
   - Therefore, the domain of $$ x $$ is $$ x \in \left[-\frac{1}{4}, \infty\right) $$.

2. **Determine the range:**
   - Since $$ y = \sqrt{4x + 1} $$, the smallest value occurs when $$ x = -\frac{1}{4} $$:
     $$
     y = \sqrt{4\left(-\frac{1}{4}\right) + 1} = \sqrt{0} = 0
     $$
   - As $$ x $$ increases, $$ 4x + 1 $$ increases, and thus $$ y $$ can take any value greater than or equal to 0.
   - Therefore, the range of $$ y $$ is $$ y \in [0, \infty) $$.

3. **Conclusion:**
   - The relation $$ y = \sqrt{4x + 1} $$ does define $$ y $$ as a function of $$ x $$ because for each $$ x $$ in the domain, there is exactly one corresponding $$ y $$.
   - The domain is $$ \left[-\frac{1}{4}, \infty\right) $$ and the range is $$ [0, \infty) $$.

Thus, the answer is:
- Does the relation define a function? **Yes**
- Domain: $$ \left[-\frac{1}{4}, \infty\right) $$
- Range: $$ [0, \infty) $$
The relation $$ y = \sqrt{4x + 1} $$ defines $$ y $$ as a function of $$ x $$. The domain is $$ x \geq -\frac{1}{4} $$ and the range is $$ y \geq 0 $$.

Decide whether the given relation defines \( y \) as a function of \( x \). Give the domain and range. \( y=\sqrt{4 x+1} \) Does the relation define a function? Yes No

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