Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 61cd

Chapter 3, Problem 61cd

Find c. (fog) (2) d. (go f) (2). f(x) = √x, g(x) = x − 1

Verified step by step guidance

Step 1: Understand the problem. You are tasked with finding two composite function values: (f ∘ g)(2) and (g ∘ f)(2). Composite functions involve substituting one function into another. The given functions are f(x) = √x and g(x) = x - 1.

Step 2: Start with (f ∘ g)(2). This means you first apply g(x) to the input 2, and then use the result as the input for f(x). Mathematically, (f ∘ g)(2) = f(g(2)).

Step 3: Calculate g(2). Substitute x = 2 into g(x) = x - 1. This gives g(2) = 2 - 1.

Step 4: Use the result of g(2) as the input for f(x). Substitute g(2) into f(x) = √x. This gives f(g(2)) = √(g(2)).

Step 5: Repeat the process for (g ∘ f)(2). This means you first apply f(x) to the input 2, and then use the result as the input for g(x). Mathematically, (g ∘ f)(2) = g(f(2)). Start by calculating f(2) = √2, and then substitute this result into g(x) = x - 1 to find g(f(2)).

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Key Concepts

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

Function Composition

Function composition involves combining two functions to create a new function. If f and g are two functions, the composition (fog)(x) means applying g first and then applying f to the result, expressed as f(g(x)). Understanding this concept is crucial for solving problems that require evaluating composite functions.

Evaluating Functions

Evaluating functions means substituting a specific input value into a function to find its output. For example, if f(x) = √x, to evaluate f(4), you would calculate √4, which equals 2. This skill is essential for determining the values of composite functions after performing the necessary compositions.

Square Root Function

The square root function, denoted as f(x) = √x, returns the non-negative value whose square is x. This function is defined only for non-negative inputs, making it important to consider the domain when evaluating or composing functions that include square roots. Understanding its properties helps in accurately solving problems involving this function.

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Related Practice

Textbook Question

Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. ƒ^-1 (1)

Textbook Question

Find a. (fog) (x) b. (go f) (x). f(x) = √x, g(x) = x − 1

Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

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Textbook Question

Use the vertical line test to identify graphs in which y is a function of x.

Textbook Question

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. r(x) = -(x + 1)²

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Textbook Question

Begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (1/2)(x − 1)²

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