Identify the range of the function $$ g(x) = -\sqrt{2x + 10} $$.

Question

UpStudy AI · Accepted Answer

To determine the **range** of the function $$ g(x) = -\sqrt{2x + 10} $$, let's follow these steps:

1. **Understand the Function:**
   - The function involves a square root, which means the expression inside the square root must be non-negative.
   - The outer negative sign affects the output of the square root.

2. **Determine the Domain:**
   - The expression inside the square root must satisfy:
     $$
     2x + 10 \geq 0 \implies x \geq -5
     $$
   - So, the **domain** of $$ g(x) $$ is $$ x \geq -5 $$.

3. **Analyze the Output $$ y = g(x) $$:**
   - Let $$ y = -\sqrt{2x + 10} $$.
   - Since square roots are always non-negative ($$ \sqrt{2x + 10} \geq 0 $$), multiplying by $$-1$$ makes $$ y $$ non-positive:
     $$
     y \leq 0
     $$
   - As $$ x $$ increases beyond $$-5$$, $$ 2x + 10 $$ increases, making $$ \sqrt{2x + 10} $$ grow larger. Consequently, $$ y = -\sqrt{2x + 10} $$ becomes more negative without bound.

4. **Conclusion on the Range:**
   - The smallest value $$ y $$ can take is $$ 0 $$ (when $$ x = -5 $$).
   - There is no lower limit to how negative $$ y $$ can be as $$ x $$ increases.
   - Therefore, the **range** of $$ g(x) $$ is all real numbers less than or equal to zero.

**Range of $$ g(x) $$:**
$$
(-\infty,\ 0\,]
$$

**Answer:**  
All real numbers y with y at most zero. In other words, the range is ( –∞, 0 ].
The range of $$ g(x) = -\sqrt{2x + 10} $$ is all real numbers less than or equal to zero, which can be written as $$ (-\infty, 0] $$.

Identify the range of the function \( g(x) = -\sqrt{2x + 10} \).

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Calculus Questions