Use the image to answer the question.

Question

Four concave curves are drawn on a coordinate plane with the x-axis from negative 6 to 6 and the y-axis from negative 4 to 4, both in increments of 2.

Given the graph of f(x)=x−1−−−−√+1
, which of the options is the graph of f(−x)
?

(1 point)
The graph of f(−x)
is option
.
Remaining Attempts : 3
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Answer 1

To find the graph of \( f(-x) \) for the function \( f(x) = \sqrt{x - 1} + 1 \), we need to understand how transformations work on functions.

1. Original function:

   • \( f(x) = \sqrt{x - 1} + 1 \)
   • This function is defined for \( x \geq 1 \) and shifts the basic square root function up by 1.

2. Transforming for \( f(-x) \):

   • To find \( f(-x) \), we substitute \( -x \) into the function:
   • \( f(-x) = \sqrt{-x - 1} + 1 \)

3. Domain of \( f(-x) \):

   • The expression inside the square root, \( -x - 1 \), must be non-negative. This means:
   • \( -x - 1 \geq 0 \)
   • \( -x \geq 1 \) or \( x \leq -1 \)
   • Therefore, this function is defined for \( x \leq -1 \).

4. Finding the graph:

   • The graph of \( f(-x) = \sqrt{-x - 1} + 1 \) will be the reflection of the graph of \( f(x) \) over the y-axis. Since \( f(x) \) starts at \( (1, 2) \), the reflection will start at \( (-1, 2) \) and then behave as a concave down curve as \( x \) decreases.

Now that we have analyzed the transformation, we need to identify the correct graph (option) that represents \( f(-x) \).

Since I cannot view images or options, please consider the characteristics of the graph we discussed:

• The graph of \( f(-x) \) will start at \( (-1, 2) \), and will be defined for \( x \leq -1 \).
• It will reflect the general shape of the concave curve of the original function \( f(x) \).

Look for the option that has a concave curve starting at \( x = -1 \) and decreases (and remains above \( y = 1 \)) as you move leftward into the negative-x territory. If you can find such a curve in your choices, that will be the correct graph for \( f(-x) \).