Question

which of the following is the graph of this square root function? f(x) = \sqrt{x - 3} + 2

Explanation:

Step1: Find the domain of the function

For the square root function \( f(x)=\sqrt{x - 3}+2 \), the expression inside the square root must be non - negative. So we solve the inequality \( x-3\geq0 \), which gives \( x\geq3 \). This means the graph of the function should start at \( x = 3 \) (or to the right of \( x=3 \)).

Step2: Find the y - intercept (when \( x = 3 \))

Substitute \( x = 3 \) into the function \( f(x)=\sqrt{x - 3}+2 \). We get \( f(3)=\sqrt{3 - 3}+2=\sqrt{0}+2 = 2 \). So the point \( (3,2) \) should be on the graph of the function.

Now let's analyze the three graphs:

• The first graph: The curve starts at \( x<0 \), which does not satisfy the domain \( x\geq3 \), so it is not the graph of \( f(x)=\sqrt{x - 3}+2 \).
• The second graph: The curve starts at \( x<0 \), which does not satisfy the domain \( x\geq3 \), so it is not the graph of \( f(x)=\sqrt{x - 3}+2 \).
• The third graph: The curve starts at \( x = 3 \) (since the domain is \( x\geq3 \)) and when \( x = 3 \), \( y=2 \), which matches our calculation. Also, as \( x \) increases, \( \sqrt{x - 3} \) increases, so \( f(x)=\sqrt{x - 3}+2 \) increases, which is consistent with the shape of the third graph.

Answer:

The third graph (the one with the grid where the curve starts at \( x = 3,y = 2 \) and increases as \( x \) increases, with \( x \) - axis values starting from - 1 and going to 9 and \( y \) - axis values from 1 to 8).