Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 39a

Chapter 3, Problem 39a

Find ƒ+g and determine the domain for each function. f(x) = √x, g(x) = x − 4

Verified step by step guidance

Step 1: Understand the problem. You are tasked with finding the sum of two functions, ƒ(x) and g(x), which is denoted as (ƒ + g)(x). This means you need to add the two functions together: (ƒ + g)(x) = ƒ(x) + g(x).

Step 2: Write the expressions for ƒ(x) and g(x). Here, ƒ(x) = √x and g(x) = x − 4. Substitute these into the formula for (ƒ + g)(x): (ƒ + g)(x) = √x + (x − 4).

Step 3: Simplify the expression if possible. In this case, the expression (ƒ + g)(x) = √x + x − 4 cannot be simplified further because the square root and linear terms are not like terms.

Step 4: Determine the domain of the resulting function. The domain is the set of all x-values for which the function is defined. For ƒ(x) = √x, the square root requires x ≥ 0 (since you cannot take the square root of a negative number in the real number system). For g(x) = x − 4, there are no restrictions on x. Therefore, the domain of (ƒ + g)(x) is x ≥ 0.

Step 5: Write the final domain in interval notation. Since x must be greater than or equal to 0, the domain is [0, ∞).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Addition

Function addition involves combining two functions, f(x) and g(x), to create a new function, denoted as (f + g)(x) = f(x) + g(x). In this case, you will add the outputs of f(x) and g(x) for each input x, resulting in a new expression that represents the sum of the two functions.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the functions f(x) = √x and g(x) = x - 4, you must consider the restrictions imposed by each function, particularly the square root function, which requires non-negative inputs.

Combining Domains

When adding two functions, the domain of the resulting function (f + g) is determined by the intersection of the individual domains of f and g. This means you need to find the common x-values that satisfy the conditions for both functions, ensuring that the combined function is valid across its entire domain.

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Combinations

Related Practice

Textbook Question

Find f−g and determine the domain for each function. f(x) = √x, g(x) = x − 4

Textbook Question

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 of f and ƒ¯¹. f(x)=2x-1

Textbook Question

Graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3

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Write the standard form of the equation of the circle with the given center and radius. Center (-4, 0), r = 10

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Textbook Question

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

Textbook Question

Find fg and determine the domain for each function. f(x) = √x, g(x) = x − 4