Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 1 - Functions

Problema 1.1.59

Capitolo 1, Problema 1.1.59

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

(g o ƒ ) (x) = x⁴ + 3

Guida verificata passo dopo passo

Step 1: Understand the composition of functions. The composition (g o ƒ)(x) means g(ƒ(x)). We need to find a function ƒ(x) such that when we apply g to ƒ(x), we get x^4 + 3.

Step 2: Analyze the given function g(x) = x^2 + 3. Notice that g(x) takes an input, squares it, and then adds 3.

Step 3: To achieve the composition (g o ƒ)(x) = x^4 + 3, we need ƒ(x) such that when squared, it results in x^4. This suggests that ƒ(x) should be a function that, when squared, gives x^4.

Step 4: Consider the function ƒ(x) = x^2. When we apply g to ƒ(x), we have g(ƒ(x)) = g(x^2) = (x^2)^2 + 3 = x^4 + 3.

Step 5: Verify that the function ƒ(x) = x^2 satisfies the composition (g o ƒ)(x) = x^4 + 3. Therefore, ƒ(x) = x^2 is the function we are looking for.

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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, we are looking for a function ƒ such that when g is applied to ƒ, the result is a new function. This is denoted as (g o ƒ)(x), which means g(ƒ(x)). Understanding how to manipulate and combine functions is crucial for solving the problem.

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Quadratic and Polynomial Functions

The function g(x) = x² + 3 is a quadratic function, which is a specific type of polynomial function characterized by its degree of 2. The composition (g o ƒ)(x) results in a polynomial of degree 4, indicating that the function ƒ must also be a polynomial that, when composed with g, yields a higher degree polynomial. Recognizing the properties of polynomial functions helps in determining the form of ƒ.

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Finding Inverse Relationships

To find the function ƒ that satisfies the composition (g o ƒ)(x) = x⁴ + 3, we can think of it as finding an inverse relationship. This involves determining what input to g will produce the desired output. By setting g(ƒ(x)) equal to x⁴ + 3, we can derive ƒ(x) by manipulating the equation, which is essential for solving the problem effectively.

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Inverse Cosine

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