Problem 72 Find $f \circ g$ and \(g \circ... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 72

Find $f \circ g$ and $g \circ f,$ where $f(x)=x^{2}+2 x$ and $g(x)=\cos x$

Short Answer

Expert verified

For $f \circ g(x) = \cos^2 x + 2 \cos x$ and for $g \circ f(x) = \cos(x^2 + 2x)$.

Step by step solution

- Understand the Composite Functions

To find composite functions, you need to substitute one function into the other. For $f \circ g$, $g(x)$ is substituted into $f(x)$. Similarly, for $g \circ f$, $f(x)$ is substituted into $g(x)$.

- Find $f \circ g(x)$

First, we substitute $g(x)$ into $f(x)$. Given that $g(x) = \cos x$ and $f(x) = x^2 + 2x$, substitute $\cos x$ for $x$ in $f(x)$: \[f(g(x)) = f(\cos x) = (\cos x)^2 + 2(\cos x)\]. Therefore, \[f \circ g(x) = \cos^2 x + 2 \cos x\].

- Find $g \circ f(x)$

Next, we substitute $f(x)$ into $g(x)$. Given that $f(x) = x^2 + 2x$ and $g(x) = \cos x$, substitute $x^2 + 2x$ for $x$ in $g(x)$: \[g(f(x)) = g(x^2 + 2x) = \cos(x^2 + 2x)\]. Therefore, \[g \circ f(x) = \cos(x^2 + 2x)\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In other words, one function's output becomes the input for another function. For example, with functions $f(x)$ and $g(x)$, the composite function $f \circ g$ represents $f(g(x))$. This means we first apply $g$ to $x$ and then apply $f$ to the result of $g(x)$. Similarly, $g \circ f$ is $g(f(x))$. Understanding function composition allows us to combine different functions and explore new relationships between variables.

Trigonometric Functions

Trigonometric functions such as $\cos(x)$, $\sin(x)$, and $\tan(x)$ are fundamental in mathematics, particularly in relation to angles and periodic phenomena. In the given exercise, $g(x)$ is $\cos(x)$. When we compose functions involving trigonometric functions, it's essential to remember their properties and identities. For instance, $\cos^2(x) + \sin^2(x) = 1$. Such identities can simplify the results of composite functions, making calculations easier and helping us understand the function behavior better.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, involving terms like $x^2$, $x$, and constants. The given function $f(x) = x^2 + 2x$ is a polynomial function. Polynomial functions are straightforward to work with due to their predictable structure. They provide a clear framework for performing algebraic operations like addition, multiplication, and substitution. When composing functions, knowing how to manipulate and substitute polynomials efficiently is crucial.

Mathematical Substitution

Mathematical substitution is key in function composition. This process involves replacing a variable with a specific value or expression. For the exercise, we substitute $g(x) = \cos(x)$ into $f(x) = x^2 + 2x$ to find $f(\cos(x))$. Conversely, we substitute $f(x) = x^2 + 2x$ into $g(x) = \cos(x)$ to find $g(x^2 + 2x)$. For example, in finding $f \circ g(x)$, we replace every instance of $x$ in $f(x)$ with $\cos(x)$, resulting in $(\cos(x))^2 + 2(\cos(x))$. This technique simplifies the process of evaluating and understanding composite functions.

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91影视

Find \(f \circ g\) and \(g \circ f,\) where \(f(x)=x^{2}+2 x\) and \(g(x)=\cos x\)

Short Answer

Step by step solution

- Understand the Composite Functions

- Find \(f \circ g(x)\)

- Find \(g \circ f(x)\)

Key Concepts

Function Composition

Trigonometric Functions

Polynomial Functions

Mathematical Substitution

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