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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 22
Chapter 3, Problem 22

For the pair of functions defined, find (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x).Give the domain of each. ƒ(x)=4x2+2x, g(x)=x2-3x+2

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First, write down the given functions: \(f(x) = 4x^2 + 2x\) and \(g(x) = x^2 - 3x + 2\).
To find \((f+g)(x)\), add the two functions: \((f+g)(x) = f(x) + g(x) = (4x^2 + 2x) + (x^2 - 3x + 2)\). Combine like terms to simplify.
To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\): \((f-g)(x) = f(x) - g(x) = (4x^2 + 2x) - (x^2 - 3x + 2)\). Be careful to distribute the minus sign and then combine like terms.
To find the product \((fg)(x)\), multiply the two functions: \((fg)(x) = f(x) imes g(x) = (4x^2 + 2x)(x^2 - 3x + 2)\). Use the distributive property (FOIL) to expand the product.
To find the quotient \((f/g)(x)\), divide \(f(x)\) by \(g(x)\): \((f/g)(x) = \frac{f(x)}{g(x)} = \frac{4x^2 + 2x}{x^2 - 3x + 2}\). The domain excludes values where \(g(x) = 0\), so solve \(x^2 - 3x + 2 = 0\) to find these excluded values.

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Key Concepts

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

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For example, (ƒ+g)(x) means adding the outputs of ƒ(x) and g(x) for each x. Understanding how to perform these operations is essential to manipulate and analyze combined functions.
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Multiplying & Dividing Functions

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 is restricted to values common to both functions and, in the case of division, excludes values that make the denominator zero.
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Domain Restrictions of Composed Functions

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number exponents with coefficients. Both ƒ(x) and g(x) are polynomials, which are defined for all real numbers, simplifying domain considerations except when division is involved.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

Determine whether each relation defines a function, and give the domain and range.

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Textbook Question

Graph each piecewise-defined function.

f(x)={6−xif x≤33if x>3f(x) =\(\begin{cases}\)6 - x & \(\text{if }\) x \(\leq\) 3 \\3 & \(\text{if }\) x > 3\(\end{cases}\)

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Textbook Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

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Textbook Question

Determine whether each relation defines a function, and give the domain and range.

Textbook Question

For the pair of functions defined, find (ƒ/g)(x). Give the domain of each. See Example 2.

ƒ(x)=2x^2-3x, g(x)=x^2-x+3

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Textbook Question

Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.