Questions: Find the equation of the rational function y=f(x), given that f has a hole at x=-8, a vertical asymptote at x=7, horizontal asymptote y=2, and cuts x-axis at x=5. f(x)=(2(x+8)(x-7))/((x+8)(x+5)) f(x)=(2(x+8)(x-5))/((x+8)(x-7)) f(x)=(2 x-10)/(x-7) f(x)=((x-8)(x-5))/(2(x-8)(x-7)) f(x)=(2(x+8)(x+5))/((x-8)(x-7)) None of the above

Find the equation of the rational function y=f(x), given that f has a hole at x=-8, a vertical asymptote at x=7, horizontal asymptote y=2, and cuts x-axis at x=5. f(x)=(2(x+8)(x-7))/((x+8)(x+5))
f(x)=(2(x+8)(x-5))/((x+8)(x-7))
f(x)=(2 x-10)/(x-7)
f(x)=((x-8)(x-5))/(2(x-8)(x-7))
f(x)=(2(x+8)(x+5))/((x-8)(x-7))

None of the above

Transcript text: Find the equation of the rational function $y=f(x)$, given that $f$ has a hole at $x=-8$, a vertical asymptote at $x=7$, horizontal asymptote $y=2$, and cuts $x$-axis at $x=5$. $f(x)=\frac{2(x+8)(x-7)}{(x+8)(x+5)}$ $f(x)=\frac{2(x+8)(x-5)}{(x+8)(x-7)}$ $f(x)=\frac{2 x-10}{x-7}$ $f(x)=\frac{(x-8)(x-5)}{2(x-8)(x-7)}$ $f(x)=\frac{2(x+8)(x+5)}{(x-8)(x-7)}$ None of the above

Solution

Solution Steps

To find the equation of the rational function $ y = f(x) $, we need to consider the given characteristics: a hole at $ x = -8 $, a vertical asymptote at $ x = 7 $, a horizontal asymptote at $ y = 2 $, and an x-intercept at $ x = 5 $. A hole at $ x = -8 $ suggests a factor of $ (x+8) $ in both the numerator and denominator. A vertical asymptote at $ x = 7 $ suggests a factor of $ (x-7) $ in the denominator. The horizontal asymptote $ y = 2 $ indicates that the leading coefficients of the numerator and denominator should have a ratio of 2. The x-intercept at $ x = 5 $ suggests a factor of $ (x-5) $ in the numerator. We will check each given function to see which one matches these conditions.

Step 1: Identify the Characteristics of the Function

We need to find a rational function $ y = f(x) $ that has:

A hole at $ x = -8 $ (implying a factor of $ (x + 8) $ in both the numerator and denominator),
A vertical asymptote at $ x = 7 $ (implying a factor of $ (x - 7) $ in the denominator),
A horizontal asymptote at $ y = 2 $ (indicating that the leading coefficients of the numerator and denominator should have a ratio of 2),
An x-intercept at $ x = 5 $ (implying a factor of $ (x - 5) $ in the numerator).

Step 2: Analyze Each Function

We will analyze the simplified forms of the given functions to see which one meets all the criteria.

Function 1: \[ f(x) = \frac{2(x - 7)}{(x + 5)} \]
- Hole: No
- Vertical Asymptote: Yes, at $ x = 7 $
- Horizontal Asymptote: $ y = 2 $
- x-intercept: No
Function 2: \[ f(x) = \frac{2(x - 5)}{(x - 7)} \]
- Hole: No
- Vertical Asymptote: Yes, at $ x = 7 $
- Horizontal Asymptote: $ y = 2 $
- x-intercept: Yes, at $ x = 5 $
Function 3: \[ f(x) = \frac{2(x - 5)}{(x - 7)} \]
- Same as Function 2, so the same characteristics apply.
Function 4: \[ f(x) = \frac{(x - 5)}{(2(x - 7))} \]
- Hole: No
- Vertical Asymptote: Yes, at $ x = 7 $
- Horizontal Asymptote: $ y = 0 $
- x-intercept: Yes, at $ x = 5 $
Function 5: \[ f(x) = \frac{2(x + 5)(x + 8)}{(x - 8)(x - 7)} \]
- Hole: No
- Vertical Asymptote: Yes, at $ x = 7 $
- Horizontal Asymptote: $ y = 0 $
- x-intercept: No

Step 3: Determine the Valid Function

From the analysis, Functions 2 and 3 are the only ones that have a vertical asymptote at $ x = 7 $, a horizontal asymptote at $ y = 2 $, and an x-intercept at $ x = 5 $. However, they do not have a hole at $ x = -8 $. Therefore, none of the provided functions meet all the criteria.