Beskou: $$ g(x)=\frac{a}{x+p}+q $$ Die volgende inligting van $$ g $$ word gegee: - Definisieversameling: $$ x \in \mathrm{R} ; x
eq-2 $$ - $$ x $$-afsnit by $$ \mathrm{K}(1 ; 0) $$ - $$ y $$-afsnit by $$ \mathrm{N}\left(0 ;-\frac{1}{2} ight) $$ 5.1 Toon dat die vergelyking van $$ g $$ geges word deur: $$ g(x)=\frac{-3}{x+2}+1 $$ 5.2 Skryf die waardeversameling van $$ g $$ neer. 5.3 Bepaal die vergelyking van $$ h $$, die simmetrie-as van $$ g $$, in die vorm $$ y=m x+c $$, waar $$ m0 $$. 5.4 Skryf die koördinate van $$ \mathrm{K}^{\prime} $$, die beeld van K gereflekteer oor $$ h $$, neer.

Question

Beskou: $$ g(x)=\frac{a}{x+p}+q $$ Die volgende inligting van $$ g $$ word gegee: - Definisieversameling: $$ x \in \mathrm{R} ; x 
eq-2 $$ - $$ x $$-afsnit by $$ \mathrm{K}(1 ; 0) $$ - $$ y $$-afsnit by $$ \mathrm{N}\left(0 ;-\frac{1}{2}ight) $$ 5.1 Toon dat die vergelyking van $$ g $$ geges word deur: $$ g(x)=\frac{-3}{x+2}+1 $$ 5.2 Skryf die waardeversameling van $$ g $$ neer. 5.3 Bepaal die vergelyking van $$ h $$, die simmetrie-as van $$ g $$, in die vorm $$ y=m x+c $$, waar $$ m>0 $$. 5.4 Skryf die koördinate van $$ \mathrm{K}^{\prime} $$, die beeld van K gereflekteer oor $$ h $$, neer.

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### 5.1 Show that the equation of $$ g $$ is given by: $$ g(x)=\frac{-3}{x+2}+1 $$

We know the general form of the function is:

$$
g(x) = \frac{a}{x+p} + q
$$

From the information provided:
- The $$ x $$-intercept is at $$ K(1, 0) $$. This means when $$ g(1) = 0 $$.
- The $$ y $$-intercept is at $$ N(0, -\frac{1}{2}) $$. This means when $$ g(0) = -\frac{1}{2} $$.
- The domain is $$ x \in \mathbb{R} $$ and $$ x \neq -2 $$, which indicates $$ p = 2 $$.

Substituting $$ p = 2 $$ into the function gives:

$$
g(x) = \frac{a}{x+2} + q
$$

Now, we will use the intercepts to find $$ a $$ and $$ q $$.

1. **Using the $$ x $$-intercept $$ K(1, 0) $$**:

$$
g(1) = 0 \implies \frac{a}{1+2} + q = 0 \implies \frac{a}{3} + q = 0 \quad (1)
$$

2. **Using the $$ y $$-intercept $$ N(0, -\frac{1}{2}) $$**:

$$
g(0) = -\frac{1}{2} \implies \frac{a}{0+2} + q = -\frac{1}{2} \implies \frac{a}{2} + q = -\frac{1}{2} \quad (2)
$$

Now we have a system of equations:

From equation (1):
$$
q = -\frac{a}{3}
$$

Substituting $$ q $$ into equation (2):

$$
\frac{a}{2} - \frac{a}{3} = -\frac{1}{2}
$$

To solve this, we need a common denominator, which is 6:

$$
\frac{3a}{6} - \frac{2a}{6} = -\frac{1}{2}
$$

This simplifies to:

$$
\frac{a}{6} = -\frac{1}{2} \implies a = -3
$$

Now substituting $$ a $$ back into equation (1) to find $$ q $$:

$$
q = -\frac{-3}{3} = 1
$$

Thus, we have:

$$
g(x) = \frac{-3}{x+2} + 1
$$

### 5.2 Write down the range of $$ g $$

The function $$ g(x) = \frac{-3}{x+2} + 1 $$ has a vertical asymptote at $$ x = -2 $$. As $$ x $$ approaches $$ -2 $$ from the left, $$ g(x) $$ approaches $$ -\infty $$, and as $$ x $$ approaches $$ -2 $$ from the right, $$ g(x) $$ approaches $$ +\infty $$.

The horizontal asymptote is $$ y = 1 $$ (as $$ x \to \infty $$ or $$ x \to -\infty $$). Therefore, the range of $$ g $$ is:

$$
\text{Range of } g: y \in \mathbb{R}, y \neq 1
$$

### 5.3 Determine the equation of $$ h $$, the axis of symmetry of $$ g $$

The function $$ g(x) $$ is a rational function and does not have a traditional axis of symmetry like a parabola. However, we can find the line of symmetry by reflecting points across the horizontal asymptote $$ y = 1 $$.

The equation of the line of symmetry $$ h $$ is:

$$
y = 1
$$

### 5.4 Write down the coordinates of $$ K' $$, the image of $$ K $$ reflected over $$ h $$

The point $$ K(1, 0) $$ is reflected over the line $$ y = 1 $$. The distance from $$ K $$ to the line $$ y = 1 $$ is:

$$
1 - 0 = 1
$$

Thus, the reflected point $$ K' $$ will be:

$$
K'(1, 1 + 1) = K'(1, 2)
$$

### Summary of Answers
- **5.1**: $$ g(x) = \frac{-3}{x+2} + 1 $$
- **5.2**: Range of $$ g $$: $$ y \in \mathbb{R}, y \neq 1 $$
- **5.3**: Equation of $$ h $$: $$ y = 1 $$
- **5.4**: Coordinates of $$ K' $$: $$ (1, 2) $$
- **5.1**: $$ g(x) = \frac{-3}{x+2} + 1 $$ - **5.2**: Range of $$ g $$: All real numbers except $$ y = 1 $$ - **5.3**: Axis of symmetry $$ h $$: $$ y = 1 $$ - **5.4**: Coordinates of $$ K' $$: $$ (1, 2) $$

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