Use the graphs of f and g to evaluate each composite function.

(fog) (-1)

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Step 1: Understand the composite function (fog)(-1). This means we first evaluate g(-1) and then use that result as the input for f(x).

Step 2: Locate the value of g(-1) on the graph of g(x) (blue curve). Find the x-coordinate of -1 and determine the corresponding y-value.

Step 3: Use the y-value obtained from g(-1) as the input for f(x). Locate this value on the graph of f(x) (red curve) and determine the corresponding y-value.

Step 4: The final y-value obtained from f(g(-1)) is the result of the composite function (fog)(-1).

Step 5: Ensure all values are correctly read from the graph and verify the steps to avoid errors in interpretation.

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Key Concepts

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

Composite Functions

A composite function is formed when one function is applied to the result of another function. It is denoted as (f ∘ g)(x), meaning f(g(x)). To evaluate a composite function, you first find the output of g for a given input, and then use that output as the input for f.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, if f(x) = 2x + 3, evaluating f(-1) means substituting -1 for x, resulting in f(-1) = 2(-1) + 3 = 1. This process is crucial for finding values in composite functions.

Graph Interpretation

Interpreting graphs of functions involves understanding how the visual representation relates to the function's behavior. For the functions f and g shown in the graph, one must identify the values of f and g at specific points, which aids in evaluating composite functions by tracing the outputs visually.

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Textbook Question

In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.

Textbook Question

Use the graphs of f and g to solve Exercises 83–90.

Graph f-g.

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Textbook Question