Problem 8 - Intermediate Algebra

Question

Fill in the blanks. The rational function \(f(x)=\frac{9 x}{x-10}\) is __ for \(x=10\) In other words, there is __ a on the domain of the function: \(x \neq 10\)

Step-by-Step Solution

Answer

The rational function is undefined for \(x=10\). There is a restriction on the domain: \(x \neq 10\).

Step 1: Identify the Form of the Rational Function

The function given is a rational function, which is a ratio of two polynomials where the denominator is not always zero. Here, the function is \(f(x) = \frac{9x}{x-10}\).

Step 2: Determine Where the Function Is Undefined

A function is undefined where its denominator is zero. For \(f(x) = \frac{9x}{x-10}\), the denominator \(x-10\) becomes zero when \(x = 10\). This means the function is undefined at \(x = 10\).

Step 3: State the Implication on the Domain

Since the function is undefined at \(x = 10\), \(x = 10\) is not included in the domain of the function. Therefore, the domain of the function is all real numbers except \(x = 10\). This is expressed as \(x eq 10\).

Step 4: Fill in the Blanks

The first blank is filled with 'undefined' because the function is not defined when \(x = 10\). The second blank is filled with 'a restriction' because there is a specific condition (restriction) that \(x\) cannot be 10.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we are referring to the complete set of possible input values (x-values) that will yield a valid output (y-value). These are all the values that you can plug into the function's formula and still get a result that makes sense. In the context of rational functions, however, we need to be cautious about what values might make the function undefined. The domain, then, becomes the set of all real numbers except the x-values that make the denominator zero, because division by zero is undefined.

• To find the domain, we usually identify and exclude these x-values.
• In our example, the function is undefined at
  \(x = 10\).
• Therefore, the domain excludes this value,
  which leads to a domain of all real numbers except \(x eq 10\).

Undefined Values

Undefined values in a function occur where certain x-values make the function impossible to calculate. For rational functions, this often happens when the denominator becomes zero. The rules of mathematics dictate that division by zero is undefined because it does not result in a finite number.

• To locate undefined values:
  Simply set the denominator equal to zero and solve for x.
• For our function \(f(x) = \frac{9x}{x-10}\), it becomes clear
  the function is undefined at \(x = 10\).
• These values are crucial as they indicate points that need to be excluded
  from the function's domain.

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents. They form the building blocks for rational functions when used in ratios where both the numerator and the denominator are polynomials.A clear understanding of polynomials is essential because:

• In rational functions, both the numerator and the denominator are often polynomials.
• Polynomials themselves are defined for all real numbers,
  but when in the denominator, they can have specific values that make the function undefined.
• Recognizing and manipulating polynomials can help identify where rational
  functions might face undefined values due to division by zero.

In our function, the denominator \(x - 10\) is a simple polynomial. When it equals zero, the rational function becomes undefined.