Question

[LOOK AT THE PICTURE]

Answer 1

The function g(x) = 2(1/(x + 1)) - 2 exhibits a vertical stretch by 2, a vertical translation down by 2, and a horizontal translation left by 1. The graph resembles a hyperbola. The domain is
`\(x \\eq -1\)`, and the range is
`\(y \\eq -2\)`.

The given function is
`\( g(x) = 2 \left((1)/(x + 1)\right) - 2 \)`.

Identify the transformations:

1. Vertical Stretch/Compression: The factor of 2 indicates a vertical stretch by a factor of 2.

2. Vertical Translation: The "-2" outside the function shifts the graph downward by 2 units.

3. Horizontal Translation: The "+1" inside the function shifts the graph left by 1 unit.

Now apply these transformations to the reference points:

- Reference Point (-1, -1):

- Vertical Stretch: 2 × (-1) = -2

- Vertical Translation: -2 - 2 = -4

- Horizontal Translation: -1 + 1 = 0

- New Point: (0, -4)

- Reference Point (1, 1):

- Vertical Stretch: 2 × (1) = 2

- Vertical Translation: 2 - 2 = 0

- Horizontal Translation: 1 + 1 = 2

- New Point: (2, 0)

Graph the transformed function by applying these transformations to the asymptotes x = 0 and y = 0 and the reference points. The graph will resemble a hyperbola in the first and third quadrants.

Domain and Range:

- Domain:
`\( x \\eq -1 \)` (to avoid division by zero)

- Range:
`\( y \\eq -2 \)` (excluding the horizontal shift downward)