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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 57ab
Chapter 3, Problem 57ab

Find a. (fog) (x) b. (go f) (x) f(x) = x²+2, g(x) = x² – 2

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Step 1: Understand the problem. You are tasked with finding two compositions of functions: (f ∘ g)(x) and (g ∘ f)(x). This means you need to substitute one function into the other. The given functions are f(x) = x² + 2 and g(x) = x² - 2.
Step 2: To find (f ∘ g)(x), substitute g(x) into f(x). This means replacing every instance of 'x' in f(x) with the expression for g(x). Start with f(x) = x² + 2, and replace 'x' with g(x) = x² - 2. The resulting expression will be f(g(x)) = ((x² - 2)²) + 2.
Step 3: Simplify the expression for f(g(x)). Expand (x² - 2)² using the formula (a - b)² = a² - 2ab + b². This gives (x² - 2)² = x⁴ - 4x² + 4. Substitute this back into f(g(x)) to get f(g(x)) = x⁴ - 4x² + 4 + 2.
Step 4: To find (g ∘ f)(x), substitute f(x) into g(x). This means replacing every instance of 'x' in g(x) with the expression for f(x). Start with g(x) = x² - 2, and replace 'x' with f(x) = x² + 2. The resulting expression will be g(f(x)) = ((x² + 2)²) - 2.
Step 5: Simplify the expression for g(f(x)). Expand (x² + 2)² using the formula (a + b)² = a² + 2ab + b². This gives (x² + 2)² = x⁴ + 4x² + 4. Substitute this back into g(f(x)) to get g(f(x)) = x⁴ + 4x² + 4 - 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(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)). Conversely, (go f)(x) means applying f first and then g, written as g(f(x)). Understanding this concept is crucial for solving the given problem.
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Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. In this case, f(x) = x² + 2 and g(x) = x² - 2 are both quadratic functions. Recognizing their structure helps in evaluating compositions and understanding their graphical representations, such as parabolas.
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Evaluating Functions

Evaluating functions involves substituting a specific value of x into the function to find the corresponding output. For example, to find (fog)(x), you first calculate g(x) and then substitute that result into f. Mastery of this process is essential for accurately computing the compositions required in the exercises.
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Related Practice
Textbook Question
In Exercises 55–59, use the graph of to graph each function g.

g(x) = -f(2x)
Textbook Question

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

Textbook Question

Find a. (fog) (2) b. (go f) (2) f(x) = x²+2, g(x) = x² – 2

Textbook Question

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

Textbook Question

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

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. (fog) (0)

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