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Chapter 0: Problem 81

Determine the domain of each function. $$f(x)=\frac{x^{3}}{x^{2}-5 x+6}$$

Short Answer

Expert verified
The domain is \[ (-\infty, 2) \cup (2, 3) \cup (3, \infty) \]

Step by step solution

01

- Identify the Function's Domain Restrictions

The domain of a function includes all real numbers, except where the function is undefined. For rational functions, the function is undefined wherever the denominator equals zero.
02

- Set the Denominator Equal to Zero

To find the values where the denominator equals zero, solve the equation: \[ x^2 - 5x + 6 = 0 \]
03

- Solve the Quadratic Equation

Factor the quadratic equation: \[ x^2 - 5x + 6 = (x - 2)(x - 3) \]Set each factor equal to zero and solve for x:\[ x - 2 = 0 \rightarrow x = 2 \]\[ x - 3 = 0 \rightarrow x = 3 \]
04

- Determine the Function's Domain

Since the function is undefined at \( x = 2 \) and \( x = 3 \), the domain of the function is all real numbers except for these values.
05

- Write the Domain in Interval Notation

Express the domain in interval notation, excluding the values where the function is undefined: \[ (-\infty, 2) \cup (2, 3) \cup (3, \infty) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational function
A rational function is a fraction where both the numerator and the denominator are polynomials. For instance, the function given in the exercise, \( f(x) = \frac{x^3}{x^2 - 5x + 6} \), has a polynomial numerator \( x^3 \) and a polynomial denominator \( x^2 - 5x + 6 \).

The domain of a rational function includes all real numbers, except where the denominator is zero because division by zero is undefined. To find these values, we solve the denominator for zero, and exclude these values from the domain.
quadratic equation
A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \). Here, the denominator of the function is a quadratic equation: \( x^2 - 5x + 6 \).

To determine where the rational function is undefined, we need to solve this quadratic equation. This can be done by factoring, completing the square, or using the quadratic formula.

For this exercise, we factor the quadratic equation:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \].

Setting each factor equal to zero gives us the roots: \( x - 2 = 0 \rightarrow x = 2 \) and \( x - 3 = 0 \rightarrow x = 3 \).
These values make the denominator zero, so we exclude them from the function's domain.
interval notation
Interval notation is a way to describe sets of numbers between endpoints. It uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints.

To represent the domain of the function \( f(x) \) in the exercise, we exclude the points where the denominator is zero (\( x = 2 \) and \( x = 3 \)). The domain in interval notation is:
\[ (-\infty, 2) \cup (2, 3) \cup (3, +\infty) \].

This format succinctly shows all real numbers that are included in the function鈥檚 domain, except for the points where the function is undefined.

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Most popular questions from this chapter

(See Exercise 68.) The Video Wizard buys a new computer system for \(\$ 60,000\) and projects that its book value will be \(\$ 2000\) after 5 yr. Using straight- line depreciation, find the book value after 3 yr.

It is theorized that the price per share of a stock is inversely proportional to the prime (interest) rate. In January 2010 , the price per share 5 of Apple Inc. stock was \(\$ 205.93\) and the prime rate \(R\) was \(3.25 \%\). The prime rate rose to \(4.75 \%\) in March 2010. (Source: finance, yahoo.com and Federal Reserve Board.) What was the price per share in March 2010 if the assumption of inverse proportionality is correct?

In 2000 the percentage of 18 - to 29 -year-olds who used the Internet was \(72 \%\). In 2009 , that percentage had risen to \(92 \%\) a) Use the year as the \(x\) -coordinate and the percentage as the \(y\) -coordinate. Find the equation of the line that contains the data points. b) Use the equation in part (a) to estimate the percentage of Internet users in 2010 . c) Use the equation in part (a) to estimate the year in which the percentage of Internet users will reach \(100 \%\) d) Explain why a linear equation cannot be used for years after the year found in part(c).

While driving a car, you see a child suddenly crossing the street. Your brain registers the emergency and sends a signal to your foot to hit the brake. The car travels a distance \(D\), in feet, during this time, where \(D\) is a function of the speed \(r,\) in miles per hour, that the car is traveling when you see the child. That reaction distance is a linear function given by \(D(r)=\frac{11 r+5}{10}\). a) Find \(D(5), D(10), D(20), D(50),\) and \(D(65).\) b) Graph \(D(r).\) c) What is the domain of the function? Explain.

At most, how many \(y\) -intercepts can a function have? Explain.
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