Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 18
Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log _{6}\left(2 x^{2}-7 x-4\right)$$
Short Answer
Expert verified
Domain: \( (-\infty, -\frac{1}{2}) \cup (4, \infty) \).
Step by step solution
01
Identify the Function
The function given is \( y = \log_{6}(2x^2 - 7x - 4) \). A logarithmic function is defined only when its argument is greater than zero.
02
Determine Restrictions on the Argument
For the function \( y = \log_{6}(2x^2 - 7x - 4) \) to be defined, the argument \( 2x^2 - 7x - 4 \) must be greater than zero. Thus, we need to solve the inequality: \( 2x^2 - 7x - 4 > 0 \).
03
Solve the Quadratic Inequality
First, solve the related quadratic equation \( 2x^2 - 7x - 4 = 0 \) to find the critical points using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a = 2 \), \( b = -7 \), \( c = -4 \). Compute the discriminant:\[ b^2 - 4ac = (-7)^2 - 4 \times 2 \times (-4) = 49 + 32 = 81 \]So, the solutions are:\[ x = \frac{7 \pm \sqrt{81}}{4} = \frac{7 \pm 9}{4} \]\( x_1 = 4 \) and \( x_2 = -\frac{1}{2} \).
04
Test Intervals for Inequality
The critical points divide the number line into intervals: \( (-\infty, -\frac{1}{2}) \), \( (-\frac{1}{2}, 4) \), and \( (4, \infty) \). Choose test points from each interval to determine where \( 2x^2 - 7x - 4 > 0 \):- For \( x = -1 \) (in \((-\infty, -\frac{1}{2})\)), \( 2(-1)^2 - 7(-1) - 4 = 2 + 7 - 4 = 5 > 0 \). Thus, this interval is part of the solution.- For \( x = 0 \) (in \((-\frac{1}{2}, 4)\)), \( 2(0)^2 - 7(0) - 4 = -4 < 0 \). Thus, this interval is not part of the solution.- For \( x = 5 \) (in \((4, \infty)\)), \( 2(5)^2 - 7(5) - 4 = 50 - 35 - 4 = 11 > 0 \). Thus, this interval is part of the solution.
05
Write the Domain
From the interval tests, the inequality \( 2x^2 - 7x - 4 > 0 \) is satisfied in the intervals \( (-\infty, -\frac{1}{2}) \) and \( (4, \infty) \). Therefore, the domain of the function is:\( (-\infty, -\frac{1}{2}) \cup (4, \infty) \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Function
A logarithmic function is an important concept in mathematics and is expressed in the form \( y = \log_b(x) \), where \( b \) is the base and must be positive, but not equal to 1. The most important highlight about the argument \( x \) is that it must always be greater than zero. This is because the logarithm of a non-positive number is undefined in the real number system. Let's break down why this is crucial:
Always ensure that the expression inside a logarithm is positive before proceeding with calculations.
- **Negative arguments**: Logarithms can't work with negative numbers as inputs in the real number system because no power of a positive base \( b \) will result in a negative number.
- **Zero argument**: Similarly, a logarithm of zero is undefined since there is no power that will transform a positive base into zero.
Always ensure that the expression inside a logarithm is positive before proceeding with calculations.
Quadratic Inequality
Quadratic inequalities involve finding the values of \( x \) for which a quadratic expression behaves in a specific way, such as being greater than or less than zero. For example, the inequality \( 2x^2 - 7x - 4 > 0 \) means we are interested in values of \( x \) that make the quadratic expression positive. Here's a quick overview:
The primary goal is to establish intervals on the number line and determine where the inequality holds.
- **Identify the inequality**: Determine the type of inequality you are dealing with (greater than, less than, etc.).
- **Solve the corresponding equation**: Start by solving the equation \( 2x^2 - 7x - 4 = 0 \) to find the points where the expression changes sign. These are called "critical points."
The primary goal is to establish intervals on the number line and determine where the inequality holds.
Interval Testing
Interval testing is a step taken after finding the critical points from solving the quadratic equation. This process helps determine where the quadratic inequality holds true across different segments of the number line. Here's how it works:
Through this method, you'll find which intervals satisfy the inequality 鈭� 2x^2 - 7x - 4 > 0. The intervals where the expression is positive become part of the domain of the function.
- **Identify intervals**: Divide the number line into different segments or intervals using the critical points. For example, if the critical points are \( x = -\frac{1}{2} \) and \( x = 4 \), the intervals are \( (-\infty, -\frac{1}{2}) \), \( (-\frac{1}{2}, 4) \), and \( (4, \infty) \).
- **Choose test points**: For each interval, choose a test point and substitute it into the quadratic expression \( 2x^2 - 7x - 4 \) to determine if the expression is positive or negative in that interval.
Through this method, you'll find which intervals satisfy the inequality 鈭� 2x^2 - 7x - 4 > 0. The intervals where the expression is positive become part of the domain of the function.
Quadratic Formula
The quadratic formula is a standardized way to find the solutions of any quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula is crucial when dealing with quadratic functions and is especially helpful in our situation for solving \( 2x^2 - 7x - 4 = 0 \). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here鈥檚 a breakdown of the components:
Using this formula, we solve the quadratic equation to find the critical values that help us test and determine the intervals satisfying the quadratic inequality.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here鈥檚 a breakdown of the components:
- \( a, b, c \) are coefficients from the quadratic equation \( ax^2 + bx + c \).
- The piece inside the square root, \( b^2 - 4ac \), is known as the "discriminant." It tells us about the nature of the solutions:
- **Positive discriminant**: Two distinct real solutions.
- **Zero discriminant**: One real, repeated solution.
- **Negative discriminant**: No real solutions, but two complex solutions.
Using this formula, we solve the quadratic equation to find the critical values that help us test and determine the intervals satisfying the quadratic inequality.
