(2) $$ y=\frac{x^{2}}{1+x} $$

To analyze the function $$ y = \frac{x^2}{1+x} $$, we can explore several aspects such as its domain, intercepts, asymptotes, and behavior. ### 1. Domain The function is defined for all $$ x $$ except where the denominator is zero. Thus, we set the denominator equal to zero: \[ 1 + x = 0 \implies x = -1 \] So, the domain of the function is $$ x \in \mathbb{R} \setminus \{-1\} $$. ### 2. Intercepts - **Y-intercept**: Set $$ x = 0 $$: \[ y = \frac{0^2}{1+0} = 0 \implies (0, 0) \] - **X-intercepts**: Set $$ y = 0 $$: \[ \frac{x^2}{1+x} = 0 \implies x^2 = 0 \implies x = 0 \implies (0, 0) \] ### 3. Asymptotes - **Vertical Asymptote**: The function has a vertical asymptote at $$ x = -1 $$ since the function approaches infinity as $$ x $$ approaches -1 from the left or right. - **Horizontal Asymptote**: To find the horizontal asymptote, we analyze the behavior of $$ y $$ as $$ x \to \infty $$ or $$ x \to -\infty $$: \[ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{x^2}{1+x} = \lim_{x \to \infty} \frac{x^2/x}{1/x + 1} = \lim_{x \to \infty} \frac{x}{1/x + 1} = \infty \] \[ \lim_{x \to -\infty} y = \lim_{x \to -\infty} \frac{x^2}{1+x} = \lim_{x \to -\infty} \frac{x^2/x}{1/x + 1} = \lim_{x \to -\infty} \frac{x}{1/x + 1} = -\infty \] ### 4. Behavior and Critical Points To find critical points, we can take the derivative of $$ y $$: Using the quotient rule: \[ y' = \frac{(1+x)(2x) - x^2(1)}{(1+x)^2} = \frac{2x + 2x^2 - x^2}{(1+x)^2} = \frac{x^2 + 2x}{(1+x)^2} = \frac{x(x + 2)}{(1+x)^2} \] Setting the derivative equal to zero to find critical points: \[ x(x + 2) = 0 \implies x = 0 \text{ or } x = -2 \] ### 5. Second Derivative Test To determine the nature of the critical points, we can use the second derivative test. We can differentiate $$ y' $$ again, but we can also analyze the sign of $$ y' $$: - For $$ x 0 $$ (increasing) - For $$ -2 0 $$, $$ y' > 0 $$ (increasing) Thus, $$ x = -2 $$ is a local maximum, and $$ x = 0 $$ is a local minimum. ### Summary - **Domain**: $$ x \in \mathbb{R} \setminus \{-1\} $$ - **Intercepts**: $$ (0, 0) $$ - **Vertical Asymptote**: $$ x = -1 $$ - **Behavior**: Local maximum at $$ x = -2 $$, local minimum at $$ x = 0 $$ - **End behavior**: $$ y \to \infty $$ as $$ x \to \infty $$ and $$ y \to -\infty $$ as $$ x \to -\infty $$ This function has interesting behavior around its critical points and asymptotes, making it a good candidate for further analysis or graphing. The function $$ y = \frac{x^2}{1+x} $$ has a domain of $$ x \in \mathbb{R} \setminus \{-1\} $$, a y-intercept at $$ (0, 0) $$, a vertical asymptote at $$ x = -1 $$, and local maxima and minima at $$ x = -2 $$ and $$ x = 0 $$, respectively. The function approaches infinity as $$ x $$ approaches positive or negative infinity.

Ask AI
Tools
- Math Solver
  
  Pre Algebra
  
  Fractions
  
  Decimals
  
  Exponents
  
  Radicals
  
  Ratios
  
  Integer
  
  Percent
  
  Algebra
  
  Equations
  
  Expressions
  
  Inequalities
  
  Functions
  
  Sequences
  
  Logarithmic
  
  Complex Numbers
  
  Matrices
  
  Pre Calculus
  
  Equations
  
  Inequalities
  
  Functions
  
  Calculus
  
  Derivatives
  
  Integrals
  
  Limits
  
  Series
  
  Trigonometry
  
  Equations
  
  Inequalities
  
  Functions
  
  Simplify
  
  Proof
  
  Geometry
  
  Statistics and Probability
  
  Matrix
- Calculators
  
  Geometry
  
  Perimeter & Area
  
  The Pythagorean Theorem
  
  Power of a Point Theorem
  
  Midpoint & Endpoint
  
  Statistic & Probability
  
  Standard Deviation
  
  Mean & Median & Mode & Range
  
  Algebra
  
  Ratio
  
  Slope
  
  Trigonometry
  
  Trigonometry Functions
  
  List of Trigonometric Identities
  
  Arithmetic
  
  Greatest Common Divisor & Least Common Multiple
  
  Percent & Decimal & Fraction
  
  Percentage
  
  Reduced Row Echelon Form (RREF)
Resources
- Question Bank
  
  Mathematics
  
  Arithmetic
  
  Pre Algebra
  
  Algebra
  
  Pre Calculus
  
  Calculus
  
  Geometry
  
  Statistics
  
  Probability
  
  Trigonometry
  
  Physics
  
  Chemistry
  
  Biology
  
  History
  
  Geography
  
  Economics
  
  English
  
  Everyday
  
  Arts
  
  Computer Technology
  
  Medicine
  
  Social Sciences
  
  Humanities
  
  Engineering
  
  Other
- Blog

Ask AI
Tools
- Math Solver
  - Pre Algebra
    - Fractions
    - Decimals
    - Exponents
    - Radicals
    - Ratios
    - Integer
    - Percent
  - Algebra
  - Pre Calculus
  - Calculus
    - Derivatives
    - Integrals
    - Limits
    - Series
  - Trigonometry
    - Equations
    - Inequalities
    - Functions
    - Simplify
    - Proof
  - Geometry
  - Statistics and Probability
  - Matrix
- Calculators
  - Geometry
  - Statistic & Probability
    - Standard Deviation
    - Mean & Median & Mode & Range
  - Algebra
    - Ratio
    - Slope
  - Trigonometry
    - Trigonometry Functions
    - List of Trigonometric Identities
  - Arithmetic
Resources
- Question Bank
  - Mathematics
    - Arithmetic
    - Pre Algebra
    - Algebra
    - Pre Calculus
    - Calculus
    - Geometry
    - Statistics
    - Probability
    - Trigonometry
  - Physics
  - Chemistry
  - Biology
  - History
  - Geography
  - Economics
  - English
  - Everyday
  - Arts
  - Computer Technology
  - Medicine
  - Social Sciences
  - Humanities
  - Engineering
  - Other
- Blog

Premium

Home Question Bank Math Algebra

Question

(2)

Ask by Hodges Murray. in China

Dec 09,2024

Upstudy AI Solution

Tutor-Verified Answer

Answer

The function

has a domain of

, a y-intercept at

, a vertical asymptote at

, and local maxima and minima at

and

, respectively. The function approaches infinity as

approaches positive or negative infinity.

Solution

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

Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Mind Expander

Did you know that the function

resembles some classic forms from calculus? It models a rational function and includes a polynomial in the numerator. This function demonstrates critical points and asymptotic behavior as

approaches -1, showcasing how rational functions can behave oddly as they approach their denominators.

Now, let’s talk about applications! You can find similar functions in various real-world scenarios. For instance, this type of equation might model the behavior of certain materials under stress, where the output is influenced by the material’s properties and the load applied. Engineers often use such functions to predict failure points or optimize designs based on load conditions.

Latest Algebra Questions

Algebra Guam Mar 31, 2025

Prev Next

(2)

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions