https://file.aiquickdraw.com/mathbot/file/1725599286692q5yob757.jpg

The function given in the image is:

\[
f(x) = 3\sqrt{-2x} + 1
\]

I will now provide information for the various required details: 

1. **Domain**: The square root function requires that the expression inside the square root must be non-negative. Hence:

\[
-2x \geq 0 \implies x \leq 0
\]

Therefore, the domain of the function is \( (-\infty, 0] \).

2. **Range**: The square root function produces non-negative values. Therefore, \( \sqrt{-2x} \geq 0 \), and multiplying by 3 gives \( 3\sqrt{-2x} \geq 0 \). So, the function starts at \( f(x) = 1 \) when \( x = 0 \), and the values increase as \( x \) decreases. Therefore, the range is \( [1, \infty) \).

3. **X-intercept**: To find the x-intercept, we set \( f(x) = 0 \):

\[
3\sqrt{-2x} + 1 = 0 \implies 3\sqrt{-2x} = -1 \quad (\text{No solution because a square root cannot be negative}).
\]

Thus, there are no x-intercepts.

4. **Y-intercept**: To find the y-intercept, evaluate the function at \( x = 0 \):

\[
f(0) = 3\sqrt{-2(0)} + 1 = 3(0) + 1 = 1
\]

So, the y-intercept is at \( (0, 1) \).

5. **End behavior**: As \( x \to -\infty \), \( -2x \to \infty \), so \( \sqrt{-2x} \to \infty \), and hence \( f(x) \to \infty \). As \( x \to 0 \), \( f(x) \to 1 \).

6. **Increasing/Decreasing intervals**: The function is increasing for \( x \in (-\infty, 0] \), since the square root function increases as its argument becomes more negative.

Would you like more details on any specific part or clarification on the graph?

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Here are 5 related questions for further exploration:

1. How would the function change if the square root term were positive?
2. Can you calculate the derivative of this function to confirm the increasing/decreasing behavior?
3. How does the transformation \( +1 \) outside the square root affect the graph?
4. What would happen if the coefficient of the square root were different?
5. How would the domain change if we replaced \( -2x \) with a different expression?

**Tip:** For functions involving square roots, always check the domain first by analyzing the expression inside the root!

Find the domain, range, intercepts, and end behavior of the function f(x) = 3√(-2x) + 1, as shown in the image.

This solution explains how to find the domain, range, intercepts, and end behavior of the function f(x) = 3√(-2x) + 1. Explore key algebraic concepts such as square root functions, increasing/decreasing intervals, and the effects of transformations.

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