Let's go step by step and analyze the function given in the image:

The function is:

$f(x) = -2\sqrt{x - 5} + 3$

Point of Origin:

The image gives the point of origin as (-5, 3), so we can check this by plugging $x = -5$ into the equation: $f(-5) = -2\sqrt{-5 - 5} + 3 = -2\sqrt{-10} + 3$ But the square root of a negative number does not result in a real number. Hence, the function is not defined for $x = -5$. There seems to be an issue with the point of origin.

Domain:

The function involves a square root, which means the expression inside the square root, $x - 5$, must be greater than or equal to zero for real numbers. Hence: $x - 5 \geq 0$ $x \geq 5$ So the domain is [5, ∞), not [-3, ∞).

Range:

The function is of the form $-2\sqrt{x - 5} + 3$, which means it will start at $f(5) = 3$ and then decrease because the square root term increases as $x$ increases, and the coefficient of $-2$ makes the overall function decrease. Thus, the range is (-∞, 3], as the function decreases towards negative infinity but never goes above 3.

y-intercept:

Since the function is not defined for $x < 5$, there is no y-intercept for this function.

Min/Max value:

The function has a maximum value at $x = 5$ because the square root starts at 0, and it increases negatively, leading to the maximum of 3 at $x = 5$.

Increasing/Decreasing Interval:

Since the square root term always increases, and with the negative coefficient, the entire function is decreasing from $x = 5$ onward. Thus, the function is decreasing on the interval [5, ∞).

Summary of Correct Answers:

• Point of Origin: Not valid for (-5, 3)
• Domain: [5, ∞)
• Range: (-∞, 3]
• y-intercept: N/A
• Min/Max value: Maximum at x = 5
• Inc/Dec Interval: Decreasing on the interval [5, ∞)

Would you like further clarification on any of these points?