Graph the piecewise function.

Question

​f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column 2 x minus 1 comma 2nd Column for x greater than minus 1 2nd Row 1st Column x plus 4 comma 2nd Column for x less than or equals minus 1 EndMatrix

Question content area bottom
Part 1
Choose the correct graph below.
A.
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x y graph
B.
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x y graph
C.
-10
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x y graph
D.
-10
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y

Answer 1

To graph the piecewise function given:

\[ f(x) = \begin{cases} 2x - 1 & \text{for } x > -1 \ x + 4 & \text{for } x \leq -1 \end{cases} \]

we need to analyze both parts of the function separately.

Step 1: Graph \( f(x) = 2x - 1 \) for \( x > -1 \)

This is a linear function with a slope of 2 and a y-intercept of -1.

1. Find the point where it would intersect the boundary \( x = -1 \): \[ f(-1) = 2(-1) - 1 = -2 - 1 = -3 \] However, this point is not included in this part of the function, so we will indicate it with an open circle when we plot it.

2. The line \( y = 2x - 1 \) will continue for values greater than \(-1\).

Step 2: Graph \( f(x) = x + 4 \) for \( x \leq -1 \)

This is another linear function that has a slope of 1 and a y-intercept of 4.

1. When \( x = -1 \): \[ f(-1) = -1 + 4 = 3 \] This point is part of this piece of the function and will be marked with a solid dot.

2. As \( x \) decreases (i.e., moves left), the line continues for values less than or equal to \(-1\).

Combining these parts

• At \( x = -1 \), the function has two parts:
  • From the second segment, it is \( 3 \) (which is included).
  • From the first segment, the value approaches \( -3 \) (which is not included).

Finalizing the Graph

Now, you should have the following characteristics for your graph:

• The line \( y = 2x - 1 \) starting from just right of \(-1\) with an open circle at \((-1, -3)\).
• The line \( y = x + 4 \) starting from the point \((-1, 3)\) and continuing to the left.

With this analysis, you can select the appropriate graph:

• **Look for a graph with:
  • An open circle at \((-1, -3)\).
  • A closed dot at \((-1, 3)\).
  • A positive slope line for \( x \) values greater than \(-1\).
  • A positive slope line intersecting at \((-1, 3)\) for \( x \) values less or equal to \(-1\).**

If you have the graphs labeled A, B, C, and D, identify based on these characteristics. If you describe the graphs, I may help more specifically to point out which one fits!