Use the given function to answer the question.

Question

Information
F of X equals x, for x is less than 0. F of X equals negative x, for x is greater than 0.

Question
Which is the graph of f of x?

Answer options with 4 options
A.	
On a coordinate plane, a v shaped graph opens up.  It has an open circle at its vertex at the origin. It has coordinates at (negative 1, 1) and (1, 1).Short description, On a coordinate plane, a v shaped graph opens up. It has an open circle at its vertex at the origin. It has coordinates at (negative 1, 1) and (1, 1).,Long description,

C.	
On a coordinate plane, a line has a positive slope and an open circle at the origin.  It has coordinates (negative 1, negative 1) and (1, 1).

B.	
A graph.Short description, A graph.,Long description,
On a coordinate plane, a v shaped graph opens down. It has an open circle at its vertex at the origin. It has coordinates at (negative 1, negative 1) and (1, negative 1).

D.	
On a coordinate plane, a line has a negative slope and an open circle at the origin.  It has coordinates (negative 1, 1), and (1, negative 1).

GPT-4o mini · Answer

To analyze the function $ f(x) $ given: - $ f(x) = x $ for $ x < 0 $, which is a line with a positive slope passing through the origin. - $ f(x) = -x $ for $ x > 0 $, which is a line with a negative slope passing through the origin. At $ x = 0 $, the function is only defined for $ x < 0 $ and $ x > 0 $, meaning there is an open circle at the origin (0,0). To summarize the behavior of the function: - For negative values of $ x $ (left side of the origin), the function behaves like a line with a slope of 1 (going up). - For positive values of $ x $ (right side of the origin), the function behaves like a line with a slope of -1 (going down). This results in a "V" shape opening upwards on the left and downwards on the right, with an open circle at the origin. Now, let's evaluate the options based on this information: **A:** Describes a V-shaped graph with an open circle at the origin and points at (-1, 1) and (1, 1). This is incorrect because for $ x > 0$, $ f(x) $ should decrease. **B:** Describes a V-shaped graph opening down with an open circle at the origin and points at (-1, -1) and (1, -1). This is incorrect as we have no part of the graph downwards. **C:** Describes a line with a positive slope for negative $ x $ and an open circle at the origin, with points at (-1, -1) and (1, 1). This is incorrect as the right side should have a negative slope. **D:** Describes a line with a negative slope and an open circle at the origin, with coordinates at (-1, 1) and (1, -1). This is consistent with $ f(x) $ on both sides. However, D looks like it’s misidentifying where 1 should be mapping on either side due to ‘negative slope.’ Thus, based on the provided function, none of the answers precisely represent $ f(x) $ accurately, which should ideally be represented visually as two lines intersecting at the origin with slopes of $ 1 $ for $ x < 0$ and $-1$ for $ x > 0$ with an open circle at (0,0). In the given options, option **C** comes closest to being correct (with respect to slopes) although it is still incorrect in representing the values correctly. To summarize, there seems no option perfectly reflecting $ f(x)$, but options **C** constitutes closest mapping at points.