11 For which values of $$ x $$ is $$ \delta $$ anction: $$ g(x)=-\frac{1}{x+2}-1 $$ 8.12. 1 Determine the equation of $$ v $$ if $$ v $$ is the reflection of $$ g $$ In the $$ y $$-axis. 8.12.2 Determine the symmetry lines of $$ v $$. 8.12.3 Sketch von a new set of axes, clearly indicating the intercepts with the axes and at least one other point. 8.12.4 Sketch and label the symmetry lines of $$ v $$. Given that $$ w $$ is obtained by translating $$ g 2 $$ units to the right and 1 unit $$ u $$ 8.13.1 State the equation of $$ w $$. 8.13.2 State the asymptotes of $$ w $$. 8.13.3 State the equations of the symmetry lines of $$ w $$. 8.13.4 Determine the shortest distance between $$ w $$ and the origin.

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### 8.12.1 Determine the equation of $$ v $$ if $$ v $$ is the reflection of $$ g $$ in the $$ y $$-axis.

The function $$ g(x) $$ is given by:

$$
g(x) = -\frac{1}{x+2} - 1
$$

To find the reflection of $$ g $$ in the $$ y $$-axis, we replace $$ x $$ with $$ -x $$:

$$
v(x) = g(-x) = -\frac{1}{-x+2} - 1
$$

Now, simplifying $$ v(x) $$:

$$
v(x) = -\frac{1}{2-x} - 1
$$

### 8.12.2 Determine the symmetry lines of $$ v $$.

To find the symmetry lines of $$ v $$, we check if $$ v(x) $$ is even or odd. A function is even if $$ v(x) = v(-x) $$ and odd if $$ v(-x) = -v(x) $$.

Calculating $$ v(-x) $$:

$$
v(-x) = -\frac{1}{2+x} - 1
$$

Now, we check if $$ v(-x) = -v(x) $$:

$$
-v(x) = \frac{1}{2-x} + 1
$$

Since $$ v(-x) \neq -v(x) $$ and $$ v(x) \neq v(-x) $$, the function $$ v $$ is neither even nor odd. Therefore, there are no symmetry lines.

### 8.12.3 Sketch $$ v $$ on a new set of axes, clearly indicating the intercepts with the axes and at least one other point.

To find the intercepts:

1. **Y-intercept**: Set $$ x = 0 $$:

$$
v(0) = -\frac{1}{2} - 1 = -\frac{3}{2}
$$

So, the y-intercept is $$ (0, -\frac{3}{2}) $$.

2. **X-intercept**: Set $$ v(x) = 0 $$:

$$
-\frac{1}{2-x} - 1 = 0
$$

Solving for $$ x $$:

$$
-\frac{1}{2-x} = 1 \implies -1 = 2 - x \implies x = 3
$$

So, the x-intercept is $$ (3, 0) $$.

3. **Another point**: Let's calculate $$ v(1) $$:

$$
v(1) = -\frac{1}{2-1} - 1 = -1 - 1 = -2
$$

So, another point is $$ (1, -2) $$.

### 8.12.4 Sketch and label the symmetry lines of $$ v $$.

Since there are no symmetry lines, we will only sketch the function $$ v $$ with the intercepts and the additional point.

### 8.13.1 State the equation of $$ w $$.

The function $$ w $$ is obtained by translating $$ g $$ 2 units to the right and 1 unit up. The transformation can be expressed as:

$$
w(x) = g(x-2) + 1
$$

Calculating $$ g(x-2) $$:

$$
g(x-2) = -\frac{1}{(x-2)+2} - 1 = -\frac{1}{x} - 1
$$

Thus,

$$
w(x) = -\frac{1}{x} - 1 + 1 = -\frac{1}{x}
$$

### 8.13.2 State the asymptotes of $$ w $$.

The function $$ w(x) = -\frac{1}{x} $$ has:

- A vertical asymptote at $$ x = 0 $$.
- A horizontal asymptote at $$ y = 0 $$.

### 8.13.3 State the equations of the symmetry lines of $$ w $$.

The function $$ w(x) = -\frac{1}{x} $$ is an odd function, so it has symmetry about the origin. Therefore, the symmetry line is:

- The line $$ y = -x $$.

### 8.13.4 Determine the shortest distance between $$ w $$ and the origin.

The shortest distance from the origin to the curve $$ w(x) = -\frac{1}{x} $$ occurs when the distance function is minimized. The distance $$ d $$ from the origin to a point $$ (x, w(x)) $$ is given by:

$$
d = \sqrt{x^2 + w(x)^2} = \sqrt{x^2 + \left(-\frac{1}{x}\right)^2} = \sqrt{x^2 + \frac{1}{x^2}}
$$

To minimize $$ d $$, we can minimize $$ d^2 $$:

$$
d^2 = x^2 + \frac{1}{x^2}
$$

Taking the derivative and setting it to zero:

$$
\frac{d}{dx}(x^2 + \frac{1}{x^2}) = 2x - \frac{2}{x^3} = 0
$$

Solving for $$ x $$:

$$
2x^4 = 2 \implies x^4 = 1 \implies x = 1 \text{ (since distance cannot be negative)}
$$

Now substituting $$ x = 1 $$ back into $$ w(x) $$:

$$
w(1) = -1
$$

Thus, the point is $$ (1, -1) $$ and the distance is:

$$
d = \sqrt{1^2 + (-1)^2} = \sqrt{2}
$$

### Summary of Results:
- **Equation of $$ v $$**: $$ v(x) = -\frac{1}{2-x} - 1 $$
- **Symmetry lines of $$ v $$**: None
- **Intercepts of $$ v $$**: $$ (0, -\frac{3}{2}), (3, 0), (1, -2) $$
- **Equation of $$ w $$**: $$ w(x) = -\frac{1}{x} $$
- **Asymptotes of $$ w $$**: $$ x = 0 $$ (vertical), $$ y = 0 $$ (horizontal)
- **Symmetry lines of $$ w $$**: Line $$ y = -x $$
- **Shortest distance from $$ w $$ to the origin**: $$ \sqrt{2} $$
**Summary of Results:** 1. **Equation of $$ v $$**: $$ v(x) = -\frac{1}{2-x} - 1 $$ 2. **Symmetry Lines of $$ v $$**: None 3. **Intercepts of $$ v $$**: $$ (0, -\frac{3}{2}), (3, 0), (1, -2) $$ 4. **Equation of $$ w $$**: $$ w(x) = -\frac{1}{x} $$ 5. **Asymptotes of $$ w $$**: $$ x = 0 $$ (vertical), $$ y = 0 $$ (horizontal) 6. **Symmetry Lines of $$ w $$**: Line $$ y = -x $$ 7. **Shortest Distance from $$ w $$ to the Origin**: $$ \sqrt{2} $$

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