Math Problem Statement

f(x)=(3x^3+2x^2)/(x^2-x)

Solution

The function given is:

f(x)=3x3+2x2x2xf(x) = \frac{3x^3 + 2x^2}{x^2 - x}

To simplify and analyze this function, let's break it down:

1. Factorization:

First, let's factor both the numerator and the denominator if possible.

  • Numerator: 3x3+2x23x^3 + 2x^2 can be factored as x2(3x+2)x^2(3x + 2).
  • Denominator: x2xx^2 - x can be factored as x(x1)x(x - 1).

So, the function can be rewritten as:

f(x)=x2(3x+2)x(x1)f(x) = \frac{x^2(3x + 2)}{x(x - 1)}

2. Simplification:

We can simplify the function by canceling out a common factor of xx from the numerator and the denominator, but note that xx should not be 0, as it would make the denominator zero.

f(x)=x(3x+2)x1forx0f(x) = \frac{x(3x + 2)}{x - 1} \quad \text{for} \quad x \neq 0

3. Domain:

The function f(x)f(x) is undefined where the denominator is zero. So, x1=0x - 1 = 0, which gives x=1x = 1. Additionally, x=0x = 0 is also excluded because we canceled xx during simplification.

Thus, the domain of f(x)f(x) is all real numbers except x=0x = 0 and x=1x = 1.

4. Final Simplified Function:

After simplification, the function is:

f(x)=3x2+2xx1f(x) = \frac{3x^2 + 2x}{x - 1}

for x0x \neq 0 and x1x \neq 1.

Would you like more details or have any questions?


Related Questions:

  1. How do you find the limits of f(x)f(x) as xx approaches 0 and 1?
  2. What are the vertical asymptotes of the function f(x)f(x)?
  3. How would you differentiate the simplified function f(x)=3x2+2xx1f(x) = \frac{3x^2 + 2x}{x - 1}?
  4. What is the behavior of f(x)f(x) as xx approaches infinity?
  5. How do you sketch the graph of the simplified function f(x)f(x)?

Tip: Always check for factors common to the numerator and denominator before simplifying a rational function.

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Math Problem Analysis

Mathematical Concepts

Rational functions
Factorization
Domain and range

Formulas

Rational function simplification

Theorems

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Suitable Grade Level

Grades 11-12