Question

The function f(x) is to be graphed on a coordinate plane.

f(x) = StartLayout Enlarged left-brace 1st row 1st column negative x, 2nd column x less-than 0 2nd row 1st column 1, 2nd column x greater-than-or-equal-to 0 EndLayout

At what point should an open circle be drawn?

(–1, 0)
(0, 0)
(0, 1)
(1, 0)

Answer 1

The function \( f(x) \) is defined piecewise as follows:

- \( f(x) = -x \) for \( x < 0 \)
- \( f(x) = 1 \) for \( x \geq 0 \)

To determine where to place an open circle (indicating that a point is not included in the graph), we need to examine the transition point at \( x = 0 \).

For \( x = 0 \):
- The value according to \( f(x) = -x \) for \( x < 0 \): As \( x \) approaches \( 0 \) from the left, \( f(x) \) approaches \( 0 \).
- The value according to \( f(x) = 1 \) for \( x \geq 0 \): At \( x = 0 \), \( f(0) = 1 \).

Since the function jumps from \( 0 \) (from the left) to \( 1 \) (at \( x = 0 \)), \( f(0) = 1 \) is included in the graph (so there is a closed dot at this point), but since \( f(x) \) is not defined for \( x<0 \) at \( x=0 \), we will place an open circle at the point \( (0, 0) \).

Hence, the open circle should be drawn at the point:

**(0, 0)**.