Domanda del libro di testo
In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 3, Problema 51
Find
a. (fog) (x)
b. (go f) (x)
c. (fog) (2)
d. (go f) (2).
f(x) = 2x, g(x) = x+7
Guida verificata passo dopo passo1
Step 1: Understand the problem. You are tasked with finding the compositions of two functions, f(x) = 2x and g(x) = x + 7. Specifically, you need to compute (f ∘ g)(x), (g ∘ f)(x), (f ∘ g)(2), and (g ∘ f)(2). The symbol ∘ represents function composition, meaning you substitute one function into the other.
Step 2: To find (f ∘ g)(x), substitute g(x) into f(x). This means replacing the input of f(x) with g(x). Start with f(x) = 2x, and replace x with g(x) = x + 7. The result is f(g(x)) = 2(x + 7). Simplify this expression.
Step 3: To find (g ∘ f)(x), substitute f(x) into g(x). This means replacing the input of g(x) with f(x). Start with g(x) = x + 7, and replace x with f(x) = 2x. The result is g(f(x)) = 2x + 7. Simplify this expression.
Step 4: To find (f ∘ g)(2), use the expression for (f ∘ g)(x) that you derived in Step 2. Substitute x = 2 into the expression f(g(x)) = 2(x + 7). Simplify the result.
Step 5: To find (g ∘ f)(2), use the expression for (g ∘ f)(x) that you derived in Step 3. Substitute x = 2 into the expression g(f(x)) = 2x + 7. Simplify the result.
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Function Composition
Function composition involves combining two functions to create a new function. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then f to the result, expressed as f(g(x)). Understanding this concept is crucial for solving problems that require evaluating composite functions.
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4:56
Function Composition
Evaluating Functions
Evaluating functions means substituting a specific value into a function to find its output. For example, if f(x) = 2x, then f(2) = 2(2) = 4. This skill is essential for calculating the values of composite functions at given points, as required in the question.
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4:26Evaluating Composed Functions
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying expressions using algebraic rules. This includes operations like addition, multiplication, and applying the distributive property. Mastery of these techniques is necessary to simplify the results of function compositions and to evaluate them accurately.
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05:09Introduction to Algebraic Expressions
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