Answer :
To graph the function [tex]\( g(x) = \frac{x^2 - 4x + 4}{x} \)[/tex], let's follow a step-by-step approach:
### Step 1: Simplify the Function
1. Simplify the expression: Before graphing, simplify the given function if possible.
[tex]\[ g(x) = \frac{x^2 - 4x + 4}{x} \][/tex]
Notice that the numerator [tex]\( x^2 - 4x + 4 \)[/tex] can be factored:
[tex]\[ x^2 - 4x + 4 = (x - 2)^2 \][/tex]
So,
[tex]\[ g(x) = \frac{(x - 2)^2}{x} \][/tex]
### Step 2: Rewrite the Simplified Expression
2. Rewrite the simplified function:
[tex]\[ g(x) = \frac{(x - 2)^2}{x} = \frac{x - 2}{x} \cdot \frac{x - 2}{1} = \left(1 - \frac{2}{x}\right) \cdot (x - 2) \][/tex]
Breaking it further:
[tex]\[ g(x) = \left(\frac{x^2 - 4x + 4}{x}\right) = \frac{x(x - 4) + 4}{x} = x - 4 + \frac{4}{x} = x - 4 + \frac{4}{x} = x -4 + \frac{5}{x} \][/tex]
### Step 3: Determine Critical Points and Asymptotes
3. Evaluate critical points and asymptotes:
- The function is undefined at [tex]\( x = 0 \)[/tex] because of division by zero.
- As [tex]\( x \rightarrow \infty \)[/tex] or [tex]\( x \rightarrow -\infty \)[/tex], the term [tex]\( \frac{4}{x} \rightarrow 0 \)[/tex], so the function behaves like [tex]\( x - 4 \)[/tex].
### Step 4: Examine Key Points
4. Calculate key points for clearer plotting:
[tex]\[ g(x) = \frac{(x-2)^2}{x} \][/tex]
For specific values of [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = \frac{(-2 - 2)^2}{-2} = \frac{16}{-2} = -8 \][/tex]
- [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = \frac{(-1 - 2)^2}{-1} = \frac{9}{-1} = -9 \][/tex]
- [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = \frac{(1 - 2)^2}{1} = \frac{1}{1} = 1 \][/tex]
- [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = \frac{(2 - 2)^2}{2} = \frac{0}{2} = 0 \][/tex]
- [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = \frac{(4 - 2)^2}{4} = \frac{4}{4} = 1 \][/tex]
### Step 5: Sketch the Graph
5. Plot the function based on simplified expression and key points:
- The simplified and key-points calculations indicate asymptotic behavior around [tex]\( x = 0 \)[/tex] and linear trend [tex]\( g(x) \rightarrow x - 4 \)[/tex] for large [tex]\( x \)[/tex].
Steps to sketch:
1. Draw the coordinate axes.
2. Identify and mark the point of discontinuity at [tex]\( x = 0 \)[/tex].
3. Plot calculated points:
- [tex]\( (-2, -8) \)[/tex]
- [tex]\( (-1, -9) \)[/tex]
- [tex]\( (1, 1) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
- [tex]\( (4, 1) \)[/tex]
4. Show behavior around [tex]\( x = 0 \)[/tex]:
- As [tex]\( x \rightarrow 0^{-}, g(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow 0^{+}, g(x) \rightarrow \infty \)[/tex].
5. Draw the curves smoothly, considering asymptotic behavior at [tex]\( x = 0 \)[/tex] and [tex]\( x \rightarrow \pm \infty \)[/tex].
### Final Graph Overview
The graph will feature:
- Vertical asymptote at [tex]\( x = 0 \)[/tex].
- A parabolic segment opening still crossing along [tex]\( x \rightarrow x - 4 \)[/tex] for large [tex]\( x \)[/tex].
### Refined Graph Defined
- [tex]\[ g(x) = \frac{(x - 2)^2}{x} \][/tex] showing descriptive changes across insightful points confirms function shape across varied domains (restricted as [tex]\( x \neq 0 \)[/tex]).
Thus, this thoroughly exception dynamic analysis completes comprehensive specifics, ensuring \( g(x) = \dots(x)...\over extended scales.
### Step 1: Simplify the Function
1. Simplify the expression: Before graphing, simplify the given function if possible.
[tex]\[ g(x) = \frac{x^2 - 4x + 4}{x} \][/tex]
Notice that the numerator [tex]\( x^2 - 4x + 4 \)[/tex] can be factored:
[tex]\[ x^2 - 4x + 4 = (x - 2)^2 \][/tex]
So,
[tex]\[ g(x) = \frac{(x - 2)^2}{x} \][/tex]
### Step 2: Rewrite the Simplified Expression
2. Rewrite the simplified function:
[tex]\[ g(x) = \frac{(x - 2)^2}{x} = \frac{x - 2}{x} \cdot \frac{x - 2}{1} = \left(1 - \frac{2}{x}\right) \cdot (x - 2) \][/tex]
Breaking it further:
[tex]\[ g(x) = \left(\frac{x^2 - 4x + 4}{x}\right) = \frac{x(x - 4) + 4}{x} = x - 4 + \frac{4}{x} = x - 4 + \frac{4}{x} = x -4 + \frac{5}{x} \][/tex]
### Step 3: Determine Critical Points and Asymptotes
3. Evaluate critical points and asymptotes:
- The function is undefined at [tex]\( x = 0 \)[/tex] because of division by zero.
- As [tex]\( x \rightarrow \infty \)[/tex] or [tex]\( x \rightarrow -\infty \)[/tex], the term [tex]\( \frac{4}{x} \rightarrow 0 \)[/tex], so the function behaves like [tex]\( x - 4 \)[/tex].
### Step 4: Examine Key Points
4. Calculate key points for clearer plotting:
[tex]\[ g(x) = \frac{(x-2)^2}{x} \][/tex]
For specific values of [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = \frac{(-2 - 2)^2}{-2} = \frac{16}{-2} = -8 \][/tex]
- [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = \frac{(-1 - 2)^2}{-1} = \frac{9}{-1} = -9 \][/tex]
- [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = \frac{(1 - 2)^2}{1} = \frac{1}{1} = 1 \][/tex]
- [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = \frac{(2 - 2)^2}{2} = \frac{0}{2} = 0 \][/tex]
- [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = \frac{(4 - 2)^2}{4} = \frac{4}{4} = 1 \][/tex]
### Step 5: Sketch the Graph
5. Plot the function based on simplified expression and key points:
- The simplified and key-points calculations indicate asymptotic behavior around [tex]\( x = 0 \)[/tex] and linear trend [tex]\( g(x) \rightarrow x - 4 \)[/tex] for large [tex]\( x \)[/tex].
Steps to sketch:
1. Draw the coordinate axes.
2. Identify and mark the point of discontinuity at [tex]\( x = 0 \)[/tex].
3. Plot calculated points:
- [tex]\( (-2, -8) \)[/tex]
- [tex]\( (-1, -9) \)[/tex]
- [tex]\( (1, 1) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
- [tex]\( (4, 1) \)[/tex]
4. Show behavior around [tex]\( x = 0 \)[/tex]:
- As [tex]\( x \rightarrow 0^{-}, g(x) \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow 0^{+}, g(x) \rightarrow \infty \)[/tex].
5. Draw the curves smoothly, considering asymptotic behavior at [tex]\( x = 0 \)[/tex] and [tex]\( x \rightarrow \pm \infty \)[/tex].
### Final Graph Overview
The graph will feature:
- Vertical asymptote at [tex]\( x = 0 \)[/tex].
- A parabolic segment opening still crossing along [tex]\( x \rightarrow x - 4 \)[/tex] for large [tex]\( x \)[/tex].
### Refined Graph Defined
- [tex]\[ g(x) = \frac{(x - 2)^2}{x} \][/tex] showing descriptive changes across insightful points confirms function shape across varied domains (restricted as [tex]\( x \neq 0 \)[/tex]).
Thus, this thoroughly exception dynamic analysis completes comprehensive specifics, ensuring \( g(x) = \dots(x)...\over extended scales.
