Given: $$ f(x)=\frac{3}{x-2}+1 $$ 10.1 Write down the equations of the asymptotes of $$ f $$. 10.2 Determine coordinates of B , the $$ x $$-intercept of $$ f $$. 10.3 Write down the domain of $$ g $$ if $$ g(x)=f(x+1) $$. 10.4 One of the axes of symmetry of $$ f $$ is an increasing function. Write down the equation of this axis of symmetry. $$ y=x-p+q $$ $$ \qquad y=x-c 2)+1 $$

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**10.1 Asymptotes**

The function is  
$$
f(x)=\frac{3}{x-2}+1.
$$

- **Vertical asymptote:** Occurs when the denominator is zero. Set  
  $$
  x-2=0\quad\Longrightarrow\quad x=2.
  $$
- **Horizontal asymptote:** As $$ x \to \pm\infty $$,  
  $$
  \frac{3}{x-2}\to 0,\quad\text{so}\quad f(x)\to 1.
  $$
  
Thus, the asymptotes are:
- Vertical: $$\displaystyle x=2$$.
- Horizontal: $$\displaystyle y=1$$.

---

**10.2 $$x$$-intercept**

The $$x$$-intercept is found by setting $$ f(x)=0 $$:
$$
\frac{3}{x-2}+1=0.
$$

Subtract 1 from both sides:
$$
\frac{3}{x-2}=-1.
$$

Multiply both sides by $$x-2$$ (noting that $$x\ne2$$):
$$
3 = -\left(x-2\right).
$$

This gives:
$$
3 = -x+2.
$$

Solve for $$x$$:
$$
-x = 3-2 \quad\Longrightarrow\quad -x=1,\quad\text{so}\quad x=-1.
$$

The coordinates of the $$x$$-intercept $$B$$ are:
$$
B=(-1,0).
$$

---

**10.3 Domain of $$g$$ where $$g(x)=f(x+1)$$**

We have:
$$
g(x)=f(x+1)=\frac{3}{(x+1)-2}+1=\frac{3}{x-1}+1.
$$

The denominator $$x-1$$ cannot be zero:
$$
x-1\ne0\quad\Longrightarrow\quad x\ne1.
$$

Thus, the domain of $$g$$ is:
$$
\{x\in \mathbb{R} : x\ne1\}.
$$

---

**10.4 Axis of symmetry**

The function $$ f(x)=\frac{3}{x-2}+1 $$ can be viewed as a hyperbola with center at the intersection of its asymptotes. The asymptotes are $$ x=2 $$ and $$ y=1 $$, so the center is at:
$$
(2,1).
$$

A hyperbola with center $$(h,k)$$ is symmetric about both the lines
$$
y-k=x-h\quad \text{and} \quad y-k=-(x-h).
$$

We are asked for the axis of symmetry that is an increasing function (slope $$1$$). This line is given by:
$$
y-1 = x-2.
$$

Solving:
$$
y = x-1.
$$

Thus, the axis of symmetry is:
$$
y=x-1.
$$
**10.1 Asymptotes** - Vertical: $$ x = 2 $$ - Horizontal: $$ y = 1 $$ **10.2 $$x$$-intercept** $$ B = (-1, 0) $$ **10.3 Domain of $$ g $$** All real numbers except $$ x = 1 $$ **10.4 Axis of symmetry** $$ y = x - 1 $$

Given:
10.1 Write down the equations of the asymptotes of .
10.2 Determine coordinates of B , the -intercept of .
10.3 Write down the domain of if .
10.4 One of the axes of symmetry of is an increasing function. Write down the equation
of this axis of symmetry.

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Given: 10.1 Write down the equations of the asymptotes of . 10.2 Determine coordinates of B , the -intercept of . 10.3 Write down the domain of if . 10.4 One of the axes of symmetry of is an increasing function. Write down the equation of this axis of symmetry.