Skip to main content
Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 2 - Functions and Graphs문제 43a
3장, 문제 43a

Find ƒ+g and determine the domain for each function.
f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

검증된 단계별 안내
1
Step 1: Understand the problem. You are tasked with finding the sum of two functions, f(x) and g(x), which are rational functions. The sum of two functions is defined as (f + g)(x) = f(x) + g(x). Additionally, you need to determine the domain of the resulting function.
Step 2: Write the expressions for f(x) and g(x). Here, f(x) = (5x + 1) / (x² - 9) and g(x) = (4x - 2) / (x² - 9). Since both functions have the same denominator, you can add the numerators directly while keeping the common denominator.
Step 3: Add the numerators of f(x) and g(x). Combine (5x + 1) and (4x - 2) to get the new numerator: (5x + 1) + (4x - 2). Simplify this expression to get 9x - 1. The resulting function is (f + g)(x) = (9x - 1) / (x² - 9).
Step 4: Determine the domain of the resulting function. The domain of a rational function excludes any values of x that make the denominator equal to zero. For the denominator x² - 9, solve the equation x² - 9 = 0. Factorize it as (x - 3)(x + 3) = 0, which gives x = 3 and x = -3. These values are excluded from the domain.
Step 5: Write the domain in interval notation. The domain of the function is all real numbers except x = 3 and x = -3. In interval notation, this is expressed as (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Function Addition

Function addition involves combining two functions, f(x) and g(x), to create a new function, ƒ+g. This is done by adding their outputs for each input x, resulting in (f+g)(x) = f(x) + g(x). Understanding this concept is crucial for solving the problem as it requires the correct application of addition to the given functions.
추천 영상:
4:46
Adding & Subtracting Functions Example 1

Domain of a Function

The domain of a function is the set of all possible input values (x) for which the function is defined. For rational functions like f(x) and g(x), the domain is restricted by values that make the denominator zero. Identifying these restrictions is essential to determine the valid inputs for the combined function ƒ+g.
추천 영상:
3:51
Domain Restrictions of Composed Functions

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this case, both f(x) and g(x) are rational functions with a common denominator (x² - 9). Understanding the properties of rational functions, including their behavior near vertical asymptotes and discontinuities, is important for analyzing the resulting function and its domain.
추천 영상:
6:04
Intro to Rational Functions
관련 실천
교과서 질문

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x − 3)² + (y + 1)² = 36

811
views
교과서 질문

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x − 1)², x ≤ 1

637
views
교과서 질문

Use the graph of y = f(x) to graph each function g. g(x) = (1/2)f(2x)

635
views
교과서 질문

Find f−g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

1195
views
교과서 질문

Find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

1289
views
교과서 질문

In Exercises 41–44, use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (4, -7) and parallel to the line whose equation is 3x + y - 9 = 0.

305
views