Answer
- **4.1**
Vertical Asymptote: \( x = 1 \)
Horizontal Asymptote: \( y = -2 \)
- **4.2**
Graph of \( f(x) = \frac{2}{x-1} - 2 \) with y-intercept at \( (0, -4) \) and x-intercept at \( (2, 0) \). Vertical asymptote at \( x = 1 \) and horizontal asymptote at \( y = -2 \) are clearly labeled.
- **4.3**
Line of symmetry: \( x = \frac{3}{2} \)
- **4.4**
\( f(x) \leq -4 \) when \( 0 \leq x < 1 \)
Solution
Let's solve the problem step by step.
### 4.1 Write down the equations of the asymptotes of \( f \).
The function given is
\[
f(x) = \frac{2}{x-1} - 2
\]
To find the asymptotes, we need to identify both vertical and horizontal asymptotes.
1. **Vertical Asymptote**: This occurs where the denominator is zero. Set the denominator equal to zero:
\[
x - 1 = 0 \implies x = 1
\]
So, the vertical asymptote is:
\[
x = 1
\]
2. **Horizontal Asymptote**: To find the horizontal asymptote, we analyze the behavior of \( f(x) \) as \( x \) approaches infinity. The term \( \frac{2}{x-1} \) approaches 0 as \( x \) approaches infinity. Therefore:
\[
f(x) \to -2 \text{ as } x \to \infty
\]
Thus, the horizontal asymptote is:
\[
y = -2
\]
### Summary of Asymptotes
- Vertical Asymptote: \( x = 1 \)
- Horizontal Asymptote: \( y = -2 \)
### 4.2 Draw a graph of \( f \).
To graph \( f(x) \), we need to find the intercepts and plot the asymptotes.
1. **Y-Intercept**: Set \( x = 0 \):
\[
f(0) = \frac{2}{0-1} - 2 = -2 - 2 = -4
\]
So, the y-intercept is \( (0, -4) \).
2. **X-Intercept**: Set \( f(x) = 0 \):
\[
\frac{2}{x-1} - 2 = 0 \implies \frac{2}{x-1} = 2 \implies 2 = 2(x-1) \implies 2 = 2x - 2 \implies 2x = 4 \implies x = 2
\]
So, the x-intercept is \( (2, 0) \).
### Graphing
- Plot the vertical asymptote \( x = 1 \) (dashed line).
- Plot the horizontal asymptote \( y = -2 \) (dashed line).
- Plot the intercepts \( (0, -4) \) and \( (2, 0) \).
The graph will approach the asymptotes and the intercepts will be clearly marked.
### 4.3 Determine the equation of the line of symmetry of \( f \) for \( m < 0 \).
The function \( f(x) \) is not symmetric about the y-axis or the x-axis. However, it can be symmetric about a vertical line. The line of symmetry can be found by determining the average of the x-coordinates of the vertical asymptote and the x-intercept.
The vertical asymptote is at \( x = 1 \) and the x-intercept is at \( x = 2 \). The line of symmetry \( x = m \) can be calculated as:
\[
m = \frac{1 + 2}{2} = \frac{3}{2}
\]
So, the equation of the line of symmetry is:
\[
x = \frac{3}{2}
\]
### 4.4 For which values of \( x \) will \( f(x) \leq -4 \)?
To find where \( f(x) \leq -4 \):
\[
\frac{2}{x-1} - 2 \leq -4
\]
Add 2 to both sides:
\[
\frac{2}{x-1} \leq -2
\]
Multiply both sides by \( x - 1 \) (note that we need to consider the sign of \( x - 1 \)):
1. **Case 1**: \( x - 1 > 0 \) (i.e., \( x > 1 \)):
\[
2 \leq -2(x - 1) \implies 2 \leq -2x + 2 \implies 2x \leq 0 \implies x \leq 0
\]
This case gives no valid solutions since \( x > 1 \).
2. **Case 2**: \( x - 1 < 0 \) (i.e., \( x < 1 \)):
\[
2 \geq -2(x - 1) \implies 2 \geq -2x + 2 \implies 2x \geq 0 \implies x \geq 0
\]
This case gives valid solutions in the interval \( 0 \leq x < 1 \).
### Final Answer for 4.4
The values of \( x \) for which \( f(x) \leq -4 \) are:
\[
0 \leq x < 1
\]
### Summary of Answers
- 4.1: Asymptotes: \( x = 1 \), \( y = -2 \)
- 4.2: Graph with intercepts \( (0, -4) \), \( (2, 0) \) and asymptotes labeled.
- 4.3: Line of symmetry: \( x = \frac{3}{2} \)
- 4.4: Values of \( x \): \( 0 \leq x < 1 \)
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