교과서 질문
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = tan(2x - π)
342
views
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.1.72
[Technology Exercise]
a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which
3/(x − 1) < 2/(x + 1)
b. Confirm your findings in part (a) algebraically.
검증된 단계별 안내1
Step 1: To graph the functions f(x) = 3/(x - 1) and g(x) = 2/(x + 1), first identify their vertical asymptotes. For f(x), the vertical asymptote is at x = 1, and for g(x), it is at x = -1. These are the values where the functions are undefined.
Step 2: Plot the graphs of both functions on the same coordinate plane. Notice that both functions are hyperbolas, and observe their behavior as x approaches the asymptotes and as x goes to positive or negative infinity.
Step 3: To find the values of x for which 3/(x - 1) < 2/(x + 1), look for the regions where the graph of f(x) is below the graph of g(x). This will give a visual indication of the solution to the inequality.
Step 4: To confirm algebraically, set up the inequality 3/(x - 1) < 2/(x + 1). Cross-multiply to eliminate the fractions, resulting in 3(x + 1) < 2(x - 1).
Step 5: Simplify the inequality from Step 4 to find the range of x values that satisfy it. This involves expanding both sides, combining like terms, and solving for x. Be sure to consider the domain restrictions due to the asymptotes.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
11m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Graphing Rational Functions
Graphing rational functions involves plotting functions that are ratios of polynomials. Key features to consider include vertical asymptotes, which occur where the denominator is zero, and horizontal asymptotes, which describe the end behavior of the function. For f(x) = 3/(x − 1) and g(x) = 2/(x + 1), vertical asymptotes are at x = 1 and x = -1, respectively.
추천 영상:
5:53
Graph of Sine and Cosine Function
Inequalities of Rational Functions
To solve inequalities involving rational functions, such as 3/(x − 1) < 2/(x + 1), one must determine where one function is less than the other. This involves finding critical points where the functions intersect or where the inequality changes sign, often requiring a combination of graphing and algebraic manipulation to identify solution intervals.
추천 영상:
6:04Intro to Rational Functions
Algebraic Confirmation of Inequalities
Confirming inequalities algebraically involves manipulating the inequality to isolate terms and solve for x. For rational functions, this often requires finding a common denominator, simplifying the inequality, and testing intervals between critical points. This process verifies the graphical solution by ensuring the inequality holds true across the identified intervals.
추천 영상:
05:25Determine Continuity Algebraically
관련 실천
교과서 질문
In Exercises 37 and 38, write a piecewise formula for the function.
<IMAGE>
206
views
교과서 질문
In Exercises 19–32, find the (a) domain and (b) range.
____
𝔂 = -2 + √1 - x
409
views
교과서 질문
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = |x| - 2
407
views
교과서 질문
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
h(t) = |t³|
214
views
교과서 질문
Evaluate cos (11π/12) as cos (π/4 + 2π/3).
238
views