List the domain and the $$ x $$-and $$ y $$-intercepts of the following function. Graph the functi Be sure to label all the asymptotes. $$ f(x)=\frac{x^{2}-7 x+6}{x+2} $$

Question

UpStudy AI · Accepted Answer

**1. Domain**

The function is  
$$
f(x)=\frac{x^2-7x+6}{x+2}.
$$
The denominator is zero when  
$$
x+2=0 \quad\Longrightarrow\quad x=-2.
$$
Thus, the domain is  
$$
\{x\in\mathbb{R} : x\neq -2\}.
$$

**2. Intercepts**

*x-intercepts* are found by setting the numerator equal to zero:
$$
x^2-7x+6=0.
$$
Factor the quadratic:
$$
x^2-7x+6=(x-1)(x-6)=0.
$$
This gives the solutions:
$$
x=1 \quad \text{and} \quad x=6.
$$
Thus, the $$x$$-intercepts are at $$(1,0)$$ and $$(6,0)$$.

*The $$y$$-intercept* is found by evaluating $$f(0)$$:
$$
f(0)=\frac{0^2-7(0)+6}{0+2}=\frac{6}{2}=3.
$$
So the $$y$$-intercept is at $$(0,3)$$.

**3. Asymptotes**

*Vertical asymptote:*  
The function is undefined when the denominator is zero (and the numerator is not zero at that point). Since $$x+2=0$$ when $$x=-2$$, there is a vertical asymptote at  
$$
x=-2.
$$

*Oblique (slant) asymptote:*  
The degree of the numerator is one more than the degree of the denominator, so perform polynomial long division of $$x^2-7x+6$$ by $$x+2$$.

- Divide $$x^2$$ by $$x$$ to get $$x$$. Multiply back: $$x(x+2)=x^2+2x$$.
- Subtract:  
  $$
  (x^2-7x+6)-(x^2+2x)=-9x+6.
  $$
- Now, divide $$-9x$$ by $$x$$ to get $$-9$$. Multiply back: $$-9(x+2)=-9x-18$$.
- Subtract:  
  $$
  (-9x+6)-(-9x-18)=24.
  $$
The division yields:
$$
f(x)=x-9+\frac{24}{x+2}.
$$
Thus, as $$x\to\pm\infty$$, the term $$\frac{24}{x+2}\to0$$ and the oblique asymptote is  
$$
y=x-9.
$$

**Summary of the Function Properties**

- **Domain:** $$ \{x\in\mathbb{R} : x\neq -2\} $$
- **$$x$$-intercepts:** $$(1,0)$$ and $$(6,0)$$
- **$$y$$-intercept:** $$(0,3)$$
- **Vertical asymptote:** $$x=-2$$
- **Oblique asymptote:** $$y=x-9$$

To graph the function, plot the intercepts, draw the vertical line at $$x=-2$$ as the vertical asymptote, and sketch the line $$y=x-9$$ as the oblique asymptote. Then, use the rational function behavior and the remainder term to graph the curve approaching the asymptotes.
**Domain:** All real numbers except $$ x = -2 $$. **Intercepts:** - $$x$$-intercepts at $$(1,0)$$ and $$(6,0)$$ - $$y$$-intercept at $$(0,3)$$ **Asymptotes:** - Vertical asymptote at $$ x = -2 $$ - Oblique asymptote at $$ y = x - 9 $$ To graph the function, plot the intercepts, draw the vertical line at $$ x = -2 $$, and sketch the line $$ y = x - 9 $$. The curve will approach these asymptotes but never touch them.

List the domain and the -and -intercepts of the following function. Graph the functi
Be sure to label all the asymptotes.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions

List the domain and the -and -intercepts of the following function. Graph the functi Be sure to label all the asymptotes.