Question
upstudy study bank question image url

Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \[ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} \] Find any oblique asymptotes of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one oblique asymptote, \( \square \) (Type an equation.) B. The function has two different oblique asymptotes. The oblique asymptote as \( x \) approaches \( -\infty \) is \( \square \) and the oblique asymptote as x approaches \( \infty \) is \( \square \) (Type equations.) C. There are no oblique asymptotes. ind the intervals where \( f(x) \) is increasing or decreasing. Select the correct choice below and fill in the answer box(es) to complete your choice. A. The function is increasing on \( \square \) It is never decreasing. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is increasing on \( \square \) It is decreasing on \( \square \) . (Type exact answers. Type your answers in interval notation. Use commas to separate answers as needed.) C. The function is decreasing on \( \square \) It is never increasing. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.)

Ask by O'Quinn Wheeler. in the United States
Mar 17,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

There are no oblique asymptotes. The function is strictly decreasing on \((-\infty,-4)\) and \((-4,\infty)\).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

To analyze the function \( f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} \), let's start with factoring the numerator and the denominator: The numerator factors to \( (x + 4)(x + 6) \) and the denominator factors to \( (x + 4)(x + 4) \), or \( (x + 4)^2 \). Therefore, we can simplify: \[ f(x) = \frac{x + 6}{x + 4} \quad \text{where } x \neq -4 \] ### Asymptotes: To find oblique asymptotes, perform polynomial long division, or note that since the degrees of the numerator and denominator are equal in the simplified form, the horizontal asymptote can be directly determined. This leads to: 1. The function has a horizontal asymptote as \( x \to \infty \) or \( x \to -\infty \) given by \( y = 1 \). So, the correct answer is: **C. There are no oblique asymptotes.** ### Increasing and Decreasing Intervals: Next, we will determine where the function is increasing or decreasing by finding the derivative: Let \( g(x) = x + 6 \) and \( h(x) = x + 4 \). By applying the quotient rule: \[ f'(x) = \frac{(g'(x)h(x) - g(x)h'(x))}{h(x)^2} = \frac{(1)(x + 4) - (x + 6)(1)}{(x + 4)^2} \] Simplifying yields: \[ f'(x) = \frac{x + 4 - x - 6}{(x + 4)^2} = \frac{-2}{(x + 4)^2} \] Since \( (x + 4)^2 > 0 \) for all \( x \neq -4 \), \( f'(x) < 0 \) everywhere except at the vertical asymptote. Thus, the function is always decreasing over the intervals \( (-\infty, -4) \) and \( (-4, \infty) \). So, the correct answer is: **B. The function is increasing on \( \emptyset \) It is decreasing on \( (-\infty, -4), (-4, \infty) \).**

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy