Which graph accurately demonstrates the relationship between the functions f(x)=x−−√

Question

and f(x)=x−−√+3
?(1 point)
Responses

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Answer 1

To analyze the relationship between the functions \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt{x} + 3 \):

1. Understand the Functions:

   • The function \( f(x) = \sqrt{x} \) is defined for \( x \geq 0 \) and its graph starts at the point (0, 0).
   • The function \( g(x) = \sqrt{x} + 3 \) is also defined for \( x \geq 0 \), but it is vertically shifted upward by 3 units compared to \( f(x) \). Hence, it starts at the point (0, 3).

2. Horizontal and Vertical Shifts:

   • Both functions will share the same x-values but the values of \( g(x) \) will always be 3 units higher than \( f(x) \) for the same value of \( x \).

3. Checking the provided options:

   • You want to find a graph where one curve (representing \( g(x) \)) is consistently 3 units above the other curve (representing \( f(x) \)) for \( x \geq 0 \).
   • The y-axis must range sufficiently to show both curves, which means it needs to reach down to at least 0 for \( f(x) \) and up to at least 3 for \( g(x) \).

Given these considerations, it is likely that the correct graph will show \( f(x) \) starting at (0,0) and \( g(x) \) starting at (0,3), with \( g(x) \) being consistently above \( f(x) \) as \( x \) increases.

The correct choice should also reflect appropriate y-axis limits, ensuring that both functions can be adequately represented.

Without seeing the actual images, I recommend you choose the graph where the y-axis includes at least from -1 or 0 to around 6 (with some space above both curves).

If the descriptions of the provided graphs suggest that one shows the range accurately covering these values, that should be your answer. Otherwise, both curves should be clear enough for you to deduce the appropriate graph.