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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 57
Capitolo 1, Problema 57

Missing piece Let g(x) = x² + 3 Find a function ƒ  that produces the given composition.


(ƒ o g) (x) = x⁴ + 6x² + 9

Guida verificata passo dopo passo
1
Step 1: Understand the composition of functions. The composition (f o g)(x) means f(g(x)). We need to find a function f such that when g(x) is plugged into f, it results in the expression x^4 + 6x^2 + 9.
Step 2: Identify g(x). We are given that g(x) = x^2 + 3. This is the inner function in the composition.
Step 3: Analyze the expression x^4 + 6x^2 + 9. Notice that this expression can be rewritten as (x^2 + 3)^2. This suggests that f(u) = u^2, where u = g(x).
Step 4: Verify the composition. Substitute g(x) = x^2 + 3 into f(u) = u^2 to check if it results in the given expression. f(g(x)) = (x^2 + 3)^2 = x^4 + 6x^2 + 9, which matches the given composition.
Step 5: Conclude that the function f(x) is x^2. Therefore, f(x) = x^2 is the function that, when composed with g(x), gives the desired result.

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Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In this case, if we have functions f and g, the composition (f o g)(x) means we first apply g to x and then apply f to the result of g. Understanding how to manipulate and derive compositions is essential for solving the problem.
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Evaluate Composite Functions - Special Cases

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c. In this problem, g(x) = x² + 3 is a quadratic function. Recognizing the properties of quadratic functions, such as their parabolic shape and how they can be transformed, is crucial for determining the appropriate function f that will yield the desired composition.
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Introduction to Polynomial Functions

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the expression. In the given composition (ƒ o g)(x) = x⁴ + 6x² + 9, the highest degree is 4, indicating that the function f must be designed to produce a polynomial of this degree when composed with g. Understanding how degrees of polynomials interact during composition helps in constructing the correct function f.
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Introduction to Polynomial Functions
Pratica correlata
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Use shifts and scalings to transform the graph of ƒ(x)=xƒ(x)=\(\sqrt{x}\)  into the graph of g. Use a graphing utility to check your work.

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Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.


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