Question

Which graph accurately demonstrates the relationship between the functions f(x)=x−−√ and f(x)=x−−√+3 ?(1 point) Responses 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. 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. 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.

Answer 1

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

1. The graph of \( f(x) = \sqrt{x} \) starts at the point (0,0) and only includes non-negative values of \( x \) because the square root function is not defined for negative \( x \).
2. The graph of \( g(x) = \sqrt{x} + 3 \) is simply the graph of \( f(x) \) shifted upward by 3 units. This means it starts at the point (0, 3) and also only exists for non-negative \( x \).

When looking for the correct graph among the provided options, you should look for the following characteristics:

- The graph of \( f(x) = \sqrt{x} \) starts at (0, 0) and rises as \( x \) increases.
- The graph of \( g(x) = \sqrt{x} + 3 \) starts at (0, 3) and rises in a similar manner, but is always 3 units above the first graph for the same \( x \).

Based on this information, identify the graph that shows:
- Two curves where one starts at (0,0) and the other starts at (0,3).
- Both curves should only appear in the first quadrant (for \( x \) values \( \geq 0 \)).

If you can identify a graph that matches this description, that will be the correct representation of the functions \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt{x} + 3 \).