Question

question 17 (2 points)

graph the inverse of the function

Explanation:

Identify key points on the original graph

The original function \(y = f(x)\) is a piecewise linear function. We can identify its key vertices and endpoints from the grid:

• Left endpoint: \((-4, 1)\)
• First turning point: \((-3, 4)\)
• Second turning point: \((-1, 4)\)
• Third turning point: \((1, 2)\)
• Right endpoint: \((4, 2)\)

Apply the inverse function property

Using the Inverse Functions concept, the graph of an inverse function \(f^{-1}(x)\) is obtained by reflecting the graph of \(f(x)\) across the line \(y = x\). This swaps the coordinates of every point: \((x, y)
ightarrow (y, x)\).

Swap coordinates for key points

We map each key point \((x, y)\) of the original function to its corresponding point \((y, x)\) on the inverse:

• \((-4, 1)

ightarrow (1, -4)\)

• \((-3, 4)

ightarrow (4, -3)\)

• \((-1, 4)

ightarrow (4, -1)\)

• \((1, 2)

ightarrow (2, 1)\)

• \((4, 2)

ightarrow (2, 4)\)

Connect the mapped points

Connect the new points in the same sequential order to form the piecewise linear segments of the inverse relation:

• Segment 1: Connect \((1, -4)\) to \((4, -3)\)
• Segment 2: Connect \((4, -3)\) to \((4, -1)\)
• Segment 3: Connect \((4, -1)\) to \((2, 1)\)
• Segment 4: Connect \((2, 1)\) to \((2, 4)\)

Answer:

To graph the inverse of the function, swap the \(x\)- and \(y\)-coordinates of the key points from the original graph and connect them in order:

• The original point \((-4, 1)\) becomes <blank>\((1, -4)\)</blank>
• The original point \((-3, 4)\) becomes <blank>\((4, -3)\)</blank>
• The original point \((-1, 4)\) becomes <blank>\((4, -1)\)</blank>
• The original point \((1, 2)\) becomes <blank>\((2, 1)\)</blank>
• The original point \((4, 2)\) becomes <blank>\((2, 4)\)</blank>

Plot these points on the coordinate plane and connect them with straight line segments in sequence.