Concept explainers
Finding the Domain of a Rational Function In Exercises 1-4, Find the domain of the function and discuss the behavior of
To calculate: The domain of the function
Answer to Problem 1RE
The domain of
Explanation of Solution
Calculation:
Consider the function
To find the domain, make the denominator zero and get the value of
As the denominator becomes zero at
Now, the behavior of
First construct the table when
For
For
Similarly, the values of
Now, construct the table when
For
Also, for
Similarly, the values of
Therefore, from the table, it is observed that as
Want to see more full solutions like this?
Chapter 4 Solutions
College Algebra Real Mathematics Real People Edition 7
- In Exercises 4-9, identify any intercepts and asymptotes of the graph of the function. Then sketch the graph of the function. f(x)=x+1x2+x12arrow_forwardIn Exercises 9-14, match the polynomial function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] f(x)=15x52x3+95xarrow_forwardIn Exercises 1-3, find the domain of the function and identify any asymptotes of the graph of the function. f(x)=3x23+x2arrow_forward
- EXAMPLE 7 Graphing a Rational Function 5x + 21 Graph r(x) and state the domain and range. x² + 10x + 25arrow_forwardEXAMPLE 5 Graphing a Rational Function 2x + 7x - 4 Graph r(x) and state the domain and range. x? + x - 2arrow_forwardEconomists use what is called a Leffer curve to predict the government revenue for tax rates from 0% to 100%. Economists agree that the end points of the curve generate 0 revenue, 10x(100-x) , where R is revenue in millions at a tax but disagree on the tax rate that produces the maximum revenue. Suppose an economist produces this rational function R(x) = 75+ x rate of x percent. Use a graphing calculator to graph the function. What tax rate produces the maximum revenue? What is the maximum revenue? OA. 39.6%; $209 million OB. 35.8%; $209 million OC. 37.5%; $210 million OD. 34.9%; $207 million Carrow_forward
- Explain, using the theorems, why the function is continuous at every number in its domain. F(x) = 2x²-x-3 x² +9 O F(x) is a polynomial, so it is continuous at every number in its domain. O F(x) is a rational function, so it is continuous at every number in its domain. O F(x) is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain. O F(x) is not continuous at every number in its domain. O none of these State the domain. (Enter your answer using interval notation.)arrow_forwardEXAMPLE 6 Graphing a Rational Function x² – 4 2x2 + 2x Graph the rational function r(x) and state the domain and range. %3Darrow_forwardHow do you determine is a function is a rational functionarrow_forward
- Limits of Rational Functions In Exercises 13-22, find the limit of each rational function (a) as x ∞ and (b) as x → ∞. Write ∞ or -∞o where appropriate. 2x + 3 13. f(x): = 5x +7 15. f(x) = x+13 x² + 14. f(x) = 2x3 +7 x3x² + x + 7 16. f(x) = 3x + 7 x² 2 - 7x3 17. h(x) = 18. h(x) = x³-3x²+6x 9x4 + x 2x45x2x+6 19. g(x) = 10x5x431 x6 x37x22 20. g(x) = x2x+1 21. f(x): = 3x7 + 5x² 6x37x+3 - 5x82x39 22. h(x) = 3+x-4x5arrow_forwardDetermine whether the statement is true or false: "The graph of a rational function cannot have both a vertical asymptote and a horizontal asymptote." If the statement is false, make the necessary change(s) to produce a true statement.arrow_forwardGive an example of a rational function.arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
- Big Ideas Math A Bridge To Success Algebra 1: Stu...AlgebraISBN:9781680331141Author:HOUGHTON MIFFLIN HARCOURTPublisher:Houghton Mifflin HarcourtCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning