Textbook Question
Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.
(ƒ o g) (x) = x⁴ + 6x² + 9
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0. Functions
Combining Functions
3:31 minutesProblem 87e
Textbook Question
Composition of even and odd functions from graphs Assume ƒ is an even function and g is an odd function. Use the (incomplete) graphs of ƒ and g in the figure to determine the following function values. <IMAGE>
e. g(g(-7))
Verified step by step guidance1
Step 1: Understand the properties of even and odd functions. An even function satisfies f(x) = f(-x) for all x, and an odd function satisfies g(x) = -g(-x) for all x.
Step 2: Identify that g is an odd function. This means that for any input x, g(x) = -g(-x).
Step 3: Evaluate g(-7) using the property of odd functions. Since g is odd, g(-7) = -g(7).
Step 4: Use the result from Step 3 to evaluate g(g(-7)). Substitute g(-7) with -g(7) in the expression g(g(-7)).
Step 5: Evaluate g(-g(7)) using the property of odd functions again. Since g is odd, g(-g(7)) = -g(g(7)).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even Functions
An even function is defined by the property that f(x) = f(-x) for all x in its domain. This symmetry about the y-axis means that the graph of an even function is mirrored on either side of the y-axis. Common examples include f(x) = x² and f(x) = cos(x). Understanding this property is crucial for analyzing the behavior of even functions in compositions.
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Odd Functions
An odd function satisfies the condition g(x) = -g(-x) for all x in its domain, indicating symmetry about the origin. This means that if you rotate the graph 180 degrees around the origin, it remains unchanged. Examples include g(x) = x³ and g(x) = sin(x). Recognizing this property is essential for evaluating compositions involving odd functions.
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Function Composition
Function composition involves applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). This process requires understanding how the output of the inner function becomes the input for the outer function. In the context of the question, evaluating g(g(-7)) necessitates first finding g(-7) and then using that result as the input for g again, highlighting the importance of sequential function evaluation.
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