Answer :
To analyze the function [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] for [tex]\( x \leq 0 \)[/tex], let's go through it step by step.
### Step 1: Define the function
We're given the function:
[tex]\[ f(x) = \frac{1}{3x - 2} \][/tex]
This function is defined for all real numbers [tex]\( x \)[/tex], except where the denominator equals zero.
### Step 2: Identify the restrictions
Identify the values of [tex]\( x \)[/tex] that make the denominator equal to zero:
[tex]\[ 3x - 2 = 0 \][/tex]
[tex]\[ 3x = 2 \][/tex]
[tex]\[ x = \frac{2}{3} \][/tex]
Since [tex]\( x = \frac{2}{3} \)[/tex] is not within our range of interest (i.e., [tex]\( x \leq 0 \)[/tex]), we don't need to worry about any undefined points in the given domain.
### Step 3: Behavior at the Boundary & Domain Analysis
For [tex]\( x \leq 0 \)[/tex]:
- As [tex]\( x \)[/tex] approaches zero from the left (close to [tex]\( -10 \)[/tex]), the expression [tex]\( 3x - 2 \)[/tex] becomes more and more negative, and hence [tex]\( \frac{1}{3x - 2} \)[/tex] becomes more and more negatively small.
- The function is continuous for all [tex]\( x \leq 0 \)[/tex] since there are no points where the function is undefined within this range.
### Step 4: Evaluating the Function
We can evaluate this function at various points to understand its behavior. Here are some evaluated points in the domain [tex]\( x \leq 0 \)[/tex]:
#### Specific Evaluations:
- [tex]\( f(-10) = \frac{1}{3(-10) - 2} = \frac{1}{-30 - 2} = \frac{1}{-32} \approx -0.03125 \)[/tex]
- [tex]\( f(-5) = \frac{1}{3(-5) - 2} = \frac{1}{-15 - 2} = \frac{1}{-17} \approx -0.05882 \)[/tex]
- [tex]\( f(-1) = \frac{1}{3(-1) - 2} = \frac{1}{-3 - 2} = \frac{1}{-5} = -0.20 \)[/tex]
- [tex]\( f(0) = \frac{1}{3(0) - 2} = \frac{1}{-2} = -0.5 \)[/tex]
#### Continuous Evaluation:
Continuing with the pattern, as we move from [tex]\( x = -10 \)[/tex] to [tex]\( x = 0 \)[/tex]:
- When [tex]\( x = -10.0, f(x) \approx -0.03125 \)[/tex]
- When [tex]\( x = -9.898, f(x) \approx -0.03154 \)[/tex]
- When [tex]\( x = -9.797, f(x) \approx -0.03185 \)[/tex]
- ...
- For values closer to zero, such as [tex]\( x = -1.0101, f(x) \approx -0.10122 \)[/tex]
- When [tex]\( x = 0.0, f(x) = -0.5 \)[/tex]
### Step 5: Conclusion on the Behavior
The function [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] shows a smooth, continuous decrease as [tex]\( x \)[/tex] moves from negative large values towards zero. Specifically, the function value [tex]\( f(x) \)[/tex] becomes increasingly negative and the magnitude grows larger. For every negative value of [tex]\( x \)[/tex], the function value is also negative but shows an increase in negative magnitude as [tex]\( x \)[/tex] approaches zero.
### Graphical Insight
If we were to graph [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] for [tex]\( x \leq 0 \)[/tex], we would expect to see a curve that starts very close to zero on the negative side for large negative [tex]\( x \)[/tex], and it becomes increasingly negative, diving sharply as [tex]\( x \)[/tex] approaches zero from the left.
This concludes the detailed, step-by-step solution for analyzing the function [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] within the given domain [tex]\( x \leq 0 \)[/tex].
### Step 1: Define the function
We're given the function:
[tex]\[ f(x) = \frac{1}{3x - 2} \][/tex]
This function is defined for all real numbers [tex]\( x \)[/tex], except where the denominator equals zero.
### Step 2: Identify the restrictions
Identify the values of [tex]\( x \)[/tex] that make the denominator equal to zero:
[tex]\[ 3x - 2 = 0 \][/tex]
[tex]\[ 3x = 2 \][/tex]
[tex]\[ x = \frac{2}{3} \][/tex]
Since [tex]\( x = \frac{2}{3} \)[/tex] is not within our range of interest (i.e., [tex]\( x \leq 0 \)[/tex]), we don't need to worry about any undefined points in the given domain.
### Step 3: Behavior at the Boundary & Domain Analysis
For [tex]\( x \leq 0 \)[/tex]:
- As [tex]\( x \)[/tex] approaches zero from the left (close to [tex]\( -10 \)[/tex]), the expression [tex]\( 3x - 2 \)[/tex] becomes more and more negative, and hence [tex]\( \frac{1}{3x - 2} \)[/tex] becomes more and more negatively small.
- The function is continuous for all [tex]\( x \leq 0 \)[/tex] since there are no points where the function is undefined within this range.
### Step 4: Evaluating the Function
We can evaluate this function at various points to understand its behavior. Here are some evaluated points in the domain [tex]\( x \leq 0 \)[/tex]:
#### Specific Evaluations:
- [tex]\( f(-10) = \frac{1}{3(-10) - 2} = \frac{1}{-30 - 2} = \frac{1}{-32} \approx -0.03125 \)[/tex]
- [tex]\( f(-5) = \frac{1}{3(-5) - 2} = \frac{1}{-15 - 2} = \frac{1}{-17} \approx -0.05882 \)[/tex]
- [tex]\( f(-1) = \frac{1}{3(-1) - 2} = \frac{1}{-3 - 2} = \frac{1}{-5} = -0.20 \)[/tex]
- [tex]\( f(0) = \frac{1}{3(0) - 2} = \frac{1}{-2} = -0.5 \)[/tex]
#### Continuous Evaluation:
Continuing with the pattern, as we move from [tex]\( x = -10 \)[/tex] to [tex]\( x = 0 \)[/tex]:
- When [tex]\( x = -10.0, f(x) \approx -0.03125 \)[/tex]
- When [tex]\( x = -9.898, f(x) \approx -0.03154 \)[/tex]
- When [tex]\( x = -9.797, f(x) \approx -0.03185 \)[/tex]
- ...
- For values closer to zero, such as [tex]\( x = -1.0101, f(x) \approx -0.10122 \)[/tex]
- When [tex]\( x = 0.0, f(x) = -0.5 \)[/tex]
### Step 5: Conclusion on the Behavior
The function [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] shows a smooth, continuous decrease as [tex]\( x \)[/tex] moves from negative large values towards zero. Specifically, the function value [tex]\( f(x) \)[/tex] becomes increasingly negative and the magnitude grows larger. For every negative value of [tex]\( x \)[/tex], the function value is also negative but shows an increase in negative magnitude as [tex]\( x \)[/tex] approaches zero.
### Graphical Insight
If we were to graph [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] for [tex]\( x \leq 0 \)[/tex], we would expect to see a curve that starts very close to zero on the negative side for large negative [tex]\( x \)[/tex], and it becomes increasingly negative, diving sharply as [tex]\( x \)[/tex] approaches zero from the left.
This concludes the detailed, step-by-step solution for analyzing the function [tex]\( f(x) = \frac{1}{3x - 2} \)[/tex] within the given domain [tex]\( x \leq 0 \)[/tex].
