Answer :
Sure, let's break down the function [tex]\( y = \frac{2x - 5}{x - 3} \)[/tex] step-by-step to understand its behavior and the steps to solve it for different purposes.
### 1. Understanding the Function:
The function given is a rational function, where both the numerator [tex]\( 2x - 5 \)[/tex] and the denominator [tex]\( x - 3 \)[/tex] are linear expressions.
### 2. Domain:
The domain of the function is all real numbers [tex]\( x \)[/tex] except where the denominator is zero, because division by zero is undefined.
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
Therefore, the domain is:
[tex]\[ x \in \mathbb{R}, x \neq 3 \][/tex]
### 3. Vertical Asymptote:
A vertical asymptote occurs where the function approaches infinity as [tex]\( x \)[/tex] approaches some value. This happens at the values excluded from the domain. Thus, the vertical asymptote is at:
[tex]\[ x = 3 \][/tex]
### 4. Horizontal Asymptote:
To find the horizontal asymptote, we look at the degrees of the numerator and the denominator.
- If the degree of the numerator (2x) is equal to the degree of the denominator (x), the horizontal asymptote is given by the ratio of the leading coefficients.
So, the leading coefficient of the numerator is 2 and the leading coefficient of the denominator is 1. Thus, the horizontal asymptote is:
[tex]\[ y = \frac{2}{1} = 2 \][/tex]
### 5. Intercepts:
#### y-intercept:
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] and solving for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2(0) - 5}{0 - 3} = \frac{-5}{-3} = \frac{5}{3} \][/tex]
#### x-intercept:
The x-intercept is found by setting [tex]\( y = 0 \)[/tex] and solving for [tex]\( x \)[/tex]. For [tex]\( \frac{2x - 5}{x - 3} = 0 \)[/tex], the numerator must be equal to zero:
[tex]\[ 2x - 5 = 0 \][/tex]
[tex]\[ 2x = 5 \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
### 6. Behavior Near Asymptotes:
- As [tex]\( x \)[/tex] approaches 3 from the left ([tex]\( x \to 3^- \)[/tex]), [tex]\( y \to -\infty \)[/tex]
- As [tex]\( x \)[/tex] approaches 3 from the right ([tex]\( x \to 3^+ \)[/tex]), [tex]\( y \to +\infty \)[/tex]
- As [tex]\( x \)[/tex] goes to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], [tex]\( y \)[/tex] approaches 2.
### Conclusion:
The function [tex]\( y = \frac{2x - 5}{x - 3} \)[/tex] can be characterized by its domain, intercepts, and asymptotic behavior:
- Domain: [tex]\( x \in \mathbb{R}, x \neq 3 \)[/tex]
- Vertical Asymptote: [tex]\( x = 3 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 2 \)[/tex]
- y-intercept: [tex]\( y = \frac{5}{3} \)[/tex]
- x-intercept: [tex]\( x = \frac{5}{2} \)[/tex]
These steps provide a comprehensive understanding of the given function.
### 1. Understanding the Function:
The function given is a rational function, where both the numerator [tex]\( 2x - 5 \)[/tex] and the denominator [tex]\( x - 3 \)[/tex] are linear expressions.
### 2. Domain:
The domain of the function is all real numbers [tex]\( x \)[/tex] except where the denominator is zero, because division by zero is undefined.
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
Therefore, the domain is:
[tex]\[ x \in \mathbb{R}, x \neq 3 \][/tex]
### 3. Vertical Asymptote:
A vertical asymptote occurs where the function approaches infinity as [tex]\( x \)[/tex] approaches some value. This happens at the values excluded from the domain. Thus, the vertical asymptote is at:
[tex]\[ x = 3 \][/tex]
### 4. Horizontal Asymptote:
To find the horizontal asymptote, we look at the degrees of the numerator and the denominator.
- If the degree of the numerator (2x) is equal to the degree of the denominator (x), the horizontal asymptote is given by the ratio of the leading coefficients.
So, the leading coefficient of the numerator is 2 and the leading coefficient of the denominator is 1. Thus, the horizontal asymptote is:
[tex]\[ y = \frac{2}{1} = 2 \][/tex]
### 5. Intercepts:
#### y-intercept:
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] and solving for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2(0) - 5}{0 - 3} = \frac{-5}{-3} = \frac{5}{3} \][/tex]
#### x-intercept:
The x-intercept is found by setting [tex]\( y = 0 \)[/tex] and solving for [tex]\( x \)[/tex]. For [tex]\( \frac{2x - 5}{x - 3} = 0 \)[/tex], the numerator must be equal to zero:
[tex]\[ 2x - 5 = 0 \][/tex]
[tex]\[ 2x = 5 \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
### 6. Behavior Near Asymptotes:
- As [tex]\( x \)[/tex] approaches 3 from the left ([tex]\( x \to 3^- \)[/tex]), [tex]\( y \to -\infty \)[/tex]
- As [tex]\( x \)[/tex] approaches 3 from the right ([tex]\( x \to 3^+ \)[/tex]), [tex]\( y \to +\infty \)[/tex]
- As [tex]\( x \)[/tex] goes to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], [tex]\( y \)[/tex] approaches 2.
### Conclusion:
The function [tex]\( y = \frac{2x - 5}{x - 3} \)[/tex] can be characterized by its domain, intercepts, and asymptotic behavior:
- Domain: [tex]\( x \in \mathbb{R}, x \neq 3 \)[/tex]
- Vertical Asymptote: [tex]\( x = 3 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 2 \)[/tex]
- y-intercept: [tex]\( y = \frac{5}{3} \)[/tex]
- x-intercept: [tex]\( x = \frac{5}{2} \)[/tex]
These steps provide a comprehensive understanding of the given function.
