Question

Continuity of a Composite Function In Exercises \(67-72\) discuss the continuity of the composite function \(h(x)=f(g(x))\) $$ \begin{array}{l}{f(x)=\sin x} \\ {g(x)=x^{2}}\end{array} $$

Short answer

The composite function \(h(x) = \sin (x^{2})\) is continuous everywhere in its domain.

Step-by-step solution

Understanding Composite Functions

A composite function is a function obtained by composing two functions. In this case, \(h(x) = f(g(x))\), is the composite function which is a result of composing \(f(x)\) and \(g(x)\). We substitute \(x\) in \(f(x)\) with \(g(x)\) which in this case is \(x^{2}\). This gives us \(h(x) = f(g(x)) = f(x^{2})\).

Substituting \(f(x)\) with given function

We know that \(f(x) = \sin x\). We substitute this in the equation derived above to get \(h(x) = \sin (x^{2})\).

Discussing the continuity of function \(h(x)\)

A function \(h(x)\) is said to be continuous at a given value of \(x\), if the limit of the function as \(x\) approaches that value, the function's value at that point, and the limit of the function as \(x\) approaches that value from the other side, are all equal. Here, since the sine function and the square function are individually continuous everywhere in their domains, their composition, which is the function \(h(x)\), will also be continuous everywhere in its domain.

Key concepts

Composite Functions

Composite functions may seem daunting at first, but they are actually a fundamental concept in mathematics that makes complex functions manageable. Whenever you see a composite function like \( h(x) = f(g(x)) \), it means you are combining two functions into one.

- **Inner Function**: The function \( g(x) \) acts as the inner function. It is applied first, calculating an intermediate value. In our case, \( g(x) = x^2 \). - **Outer Function**: The function \( f(x) \) is applied to the result of \( g(x) \). Here, \( f(x) = \sin x \).

For \( h(x) \), you first calculate \( g(x) \), which is \( x^2 \). Then, you plug it into the sine function, yielding \( h(x) = \sin(x^2) \). Understanding this layering process of functions forms the essence of composite functions.

Sine Function

The sine function, denoted by \( \sin x \), is a well-known trigonometric function. It oscillates smoothly between -1 and 1, repeating every \(2\pi\). It's what gives sine waves their shape. - **Properties**: - Sine is periodic with a period of \(2\pi\). - It is an odd function, meaning \( \sin(-x) = -\sin(x) \).- **Range**: All outputs (\( y \)) of \( \sin x \) lie within the interval [-1, 1].

In the context of our composite function \( h(x) = \sin(x^2) \), the sine function transforms the squared values, maintaining its periodic properties and defining the range of \( h(x) \) from -1 to 1, just like \( \sin x \).

Continuity

Continuity is a fundamental property in calculus that describes a function being smooth or without breaks. A function is continuous at a point \( x = a \) if:- The limit of the function as \( x \) approaches \( a \) is equal to the function's value at \( a \).

Both \( f(x) = \sin x \) and \( g(x) = x^2 \) are continuous across all real numbers. This means there are no abrupt jumps, holes, or breaks in their graphs.

The beauty of composition is that if two functions are continuous throughout their respective domains, their composite will also be continuous. This means \( h(x) = \sin(x^2) \) is continuous everywhere that \( x^2 \) is defined, and since \( x^2 \) is defined for all \( x \), \( h(x) \) is continuous everywhere.

Composition of Functions

The composition of functions is a process where one function is applied to the results of another. This operation is denoted as \( f(g(x)) \) and creates a new function by substituting the entire \( g(x) \) into \( f(x) \).- **Order Matters**: Composition is not commutative. This means \( f(g(x)) \) is generally different from \( g(f(x)) \).- **Example**: For \( h(x) = \sin(x^2) \), you see the composition by plugging \( g(x) = x^2 \) into \( f(x) = \sin x \).

This process allows complex transformations based on a sequence of basic functions, creating versatile mathematical models applicable in real-world scenarios like signal processing and oscillation modeling.