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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 47
Chapter 3, Problem 47

Find ƒ+g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

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Identify the given functions: \(f(x) = \sqrt{x + 4}\) and \(g(x) = \sqrt{x - 1}\).
Find the sum of the functions by adding them: \((f + g)(x) = f(x) + g(x) = \sqrt{x + 4} + \sqrt{x - 1}\).
Determine the domain of \(f(x)\) by setting the expression inside the square root to be greater than or equal to zero: \(x + 4 \geq 0\) which simplifies to \(x \geq -4\).
Determine the domain of \(g(x)\) similarly: \(x - 1 \geq 0\) which simplifies to \(x \geq 1\).
Find the domain of \((f + g)(x)\) by taking the intersection of the domains of \(f(x)\) and \(g(x)\), which is all \(x\) values satisfying both \(x \geq -4\) and \(x \geq 1\). This means the domain is \(x \geq 1\).

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Key Concepts

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

Function Addition (ƒ + g)

Adding two functions involves creating a new function where each output is the sum of the outputs of the original functions at the same input. For functions f and g, (ƒ + g)(x) = f(x) + g(x). This operation combines the values of both functions pointwise.
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Function Composition

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the domains of the original functions, ensuring all expressions are valid.
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Domain Restrictions of Composed Functions

Square Root Function Domain Restrictions

A square root function √(expression) is defined only when the expression inside the root is non-negative. For f(x) = √(x + 4), x + 4 ≥ 0; for g(x) = √(x − 1), x − 1 ≥ 0. These inequalities determine the domain restrictions for each function.
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Domain Restrictions of Composed Functions
Related Practice
Textbook Question

In Exercises 39–50, 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) = x³, g(x) = x³ +2

Textbook Question

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 + 2)² + (y + 2)² = 4

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Textbook Question

In Exercises 39–50, 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| +1

Textbook Question

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

Textbook Question

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = 9x/(x - 4), g(x) = 7/(x+8)

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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 off and ƒ¯¹. f(x) = (x+2)³