Problem 53 - Intermediate Algebra

Question

Simplify each function. List any restrictions on the domain. $$ f(x)=\frac{9 x^{2}+81 x}{x^{3}+9 x^{2}} $$

Step-by-Step Solution

Answer

The simplified function is \(f(x) = \frac{9}{x}\), with domain restrictions at \(x = 0\) and \(x = -9\).

Step 1: Factor the numerator

The numerator of the function is \(9x^2 + 81x\). To factor, take out the greatest common factor, which is \(9x\). This gives \(9x(x + 9)\).

Step 2: Factor the denominator

The denominator of the function is \(x^3 + 9x^2\). Similarly, factor out the greatest common factor, which is \(x^2\). This leaves \(x^2(x + 9)\).

Step 3: Simplify the expression

Now the function looks like \(\frac{9x(x + 9)}{x^2(x + 9)}\). Cancel the common factor \((x + 9)\) from both the numerator and the denominator. Be cautious that \(x + 9 eq 0\) to avoid division by zero. The simplified function is \(\frac{9x}{x^2}\).

Step 4: Further simplify the expression

Divide the numerator by the denominator: \(\frac{9x}{x^2} = \frac{9}{x}\).

Step 5: State any restrictions on the domain

The original denominator \(x^3 + 9x^2\) implies \(x^2(x + 9)\). Set each factor equal to zero to find restrictions: \(x^2 = 0\) means \(x = 0\), and \(x + 9 = 0\) means \(x = -9\). Therefore, the function is undefined at \(x = 0\) and \(x = -9\).

Key Concepts

Simplifying Functions

Simplifying functions is about reducing them to their most basic form. This is often necessary to make solving and interpreting them easier. For rational functions, like the one in our exercise, this often involves a process of canceling out common factors in the numerator and denominator. Consider the function given: \( f(x) = \frac{9x^2 + 81x}{x^3 + 9x^2} \).

• The goal is to express this function in its simplest form.
• Start by factoring both the numerator and the denominator.
• This means identifying any common algebraic components that can be simplified.

Ultimately, after simplifying, you get to a much more manageable form: \( \frac{9}{x} \). This not only makes calculations more straightforward but also reveals the nature of the function more clearly.

Domain Restrictions

In mathematics, the domain of a function is all the possible input values (usually \( x \)) for which the function is defined. For rational functions, the focus is largely on avoiding division by zero, as division by zero is undefined. Let's break this down for the exercise:

• Initially, identify the denominator, which is \( x^3 + 9x^2 \).
• Factor it to get \( x^2(x + 9) \).
• To avoid division by zero, set each factor in the denominator equal to zero and solve for \( x \).

For our function, this means:

• \( x^2 = 0 \to x = 0 \)
• \( x + 9 = 0 \to x = -9 \)

Thus, the function is not defined at \( x = 0 \) and \( x = -9 \), because these values make the original denominator zero, thereby restricting the domain.

Factoring Polynomials

Factoring is a key technique in simplifying functions and solving equations. It involves breaking down a complex expression into simpler multiplicative components or factors.In the exercise, both the numerator \( 9x^2 + 81x \) and the denominator \( x^3 + 9x^2 \) of the function needed to be factored:

• For the numerator:
  Take out the greatest common factor (GCF), \( 9x \), to factor \( 9x(x + 9) \).
• For the denominator:
  Similarly, take the GCF \( x^2 \), leading to \( x^2(x + 9) \).

Factoring transforms the function into a form where common terms in the numerator and the denominator can be canceled. This simplification process not only reduces the complexity of the function but also helps identify domain restrictions.