Problem 90
Question
$$ \text { If } f(x)=\sin x \text { and } g(x)=x^{2}, \text { then find } f \circ g(x) \text { and } \operatorname{gof}(x) \text { . } $$
Step-by-Step Solution
Verified Answer
The compositions of the functions are:
\(f \circ g(x) = \sin(x^2)\)
\(g \circ f(x) = (\sin x)^2\)
1Step 1: Find \(f \circ g(x)\)
To find \(f \circ g(x)\), we need to substitute the function \(g(x)\) in place of \(x\) in the function \(f(x)\). So, we have:
\(f \circ g(x) = f(g(x)) = \sin(g(x))\)
Now, we substitute the expression for \(g(x)\) inside this function:
\(f \circ g(x) = \sin(x^2)\)
So, \(f \circ g(x) = \sin(x^2)\).
2Step 2: Find \(g \circ f(x)\)
To find \(g \circ f(x)\), we need to substitute the function \(f(x)\) in place of \(x\) in the function \(g(x)\). So, we have:
\(g \circ f(x) = g(f(x)) = (f(x))^2\)
Now, we substitute the expression for \(f(x)\) inside this function:
\(g \circ f(x) = (\sin x)^2\)
So, \(g \circ f(x) = (\sin x)^2\).
3Step 3: Final Answer
The composition of the functions is as follows:
\(f \circ g(x) = \sin(x^2)\)
\(g \circ f(x) = (\sin x)^2\)
Key Concepts
Trigonometric FunctionsPolynomial FunctionsFunction Substitution
Trigonometric Functions
Trigonometric functions are a class of functions that relate angles of a triangle to the lengths of its sides. These functions are fundamental in studying periodic phenomena such as sound and light waves. The most common trigonometric functions include sine, cosine, and tangent. The sine function, denoted as \( \sin x \), is one of these basic functions and describes the relationship of an angle in a right triangle to its opposite side and hypotenuse.
For the particular problem at hand, \( f(x) = \sin x \) depicts the sine function. When we compose this function with another function, like a polynomial, we alter its input value. It is essential to understand that trigonometric functions are periodic, which means their values repeat at regular intervals.
For the particular problem at hand, \( f(x) = \sin x \) depicts the sine function. When we compose this function with another function, like a polynomial, we alter its input value. It is essential to understand that trigonometric functions are periodic, which means their values repeat at regular intervals.
- Sine returns values between -1 and 1.
- It has a period of \( 2\pi \).
- The key characteristic of the sine function is its smooth, wave-like form.
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are among the simplest types of functions and form the foundation for many algebraic concepts. A basic example of a polynomial function is \( g(x) = x^2 \), which is a quadratic function.
Quadratic functions have various characteristics: they produce parabolas when graphed and feature coefficients that determine their shape and direction.
Quadratic functions have various characteristics: they produce parabolas when graphed and feature coefficients that determine their shape and direction.
- The general form of a polynomial is \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \).
- Each term in the polynomial is a combination of the variable raised to a power and a coefficient.
- The degree of a polynomial is determined by the highest power of the variable.
Function Substitution
Function substitution is a technique used to simplify expressions or solve equations by replacing variables or expressions with a given function. This method is especially useful in calculus and algebra when dealing with complex equations or function compositions.
In function composition, function substitution involves replacing a variable with another function. For instance, if we have \( f(x) = \sin x \) and \( g(x) = x^2 \), and we want to find \( f \circ g(x) \), we replace the \( x \) in \( f(x) \) with \( g(x) \), leading to \( \sin(x^2) \). This is a simple example of how substitution allows one function to be nested within another, creating a new function.
In function composition, function substitution involves replacing a variable with another function. For instance, if we have \( f(x) = \sin x \) and \( g(x) = x^2 \), and we want to find \( f \circ g(x) \), we replace the \( x \) in \( f(x) \) with \( g(x) \), leading to \( \sin(x^2) \). This is a simple example of how substitution allows one function to be nested within another, creating a new function.
- The substitution method necessitates clear understanding of each function's domain and range.
- It's crucial to pay attention to the order of substitution.
- Function substitution is often used to simplify complex expressions or to integrate nested functions in calculus.
Other exercises in this chapter
Problem 88
$$ \text { If } f(x)=x^{2}-3 x+2, \text { find } f(f(x)) $$
View solution Problem 89
$$ \text { Find } \phi(\psi(x)) \text { and } \psi(\phi(x)) \text { if } \phi(x)=x^{2}+1 \text { and } \psi(x)=3^{x} \text { . } $$
View solution Problem 91
$$ \text { If } f(x)=x \cos x \text { and } g(x)=\frac{x}{1+x^{2}} \text { , then find } f \circ g(x) \text { and } g o f(x) $$
View solution Problem 92
$$ \text { If } f(x)=x^{2}-\frac{1}{x^{2}}, \text { prove that } f(x)=-f\left(\frac{1}{x}\right) \text { . } $$
View solution