Problem 32 - Calculus: One and Several Variables

Question

Form the composition \(f \circ g \circ h\) and give the domain. $$f(x)=x-1, \quad g(x)=4 x, \quad h(x)=x^{2}$$

Step-by-Step Solution

Answer

The composition \(f \circ g \circ h(x) = 4x^2 - 1\) is defined, with the domain being all real numbers (\(-\infty< x <+\infty).\)

Step 1: Evaluate Innermost Function

Begin with the innermost function \(h(x)\). As it's given \(h(x) = x^2\), there are no restrictions. Thus, the function is defined for all real numbers. Hence the domain of \(h(x)\) is \(-\infty < x < +\infty.\)

Step 2: Form and Evaluate the Composition \(g \circ h(x)\)

Now move to the next inner function, \(g(x)\), to form the composition \(g(h(x))\). This will involve substituting \(h(x)\) in place of \(x\) in the function \(g(x)\): \(g(h(x)) = g(x^{2}) =4x^{2}.\) As there are no restrictions, the composition is defined for all real numbers. Hence the domain of \(g(h(x))\) is \(-\infty< x <+\infty.\)

Step 3: Form and Evaluate the Composition \(f \circ g \circ h(x)\)

Moving to the next outer function \(f(x)\), form the composition \(f(g(h(x)))\). This will involve substituting \(g(h(x))\) instead of \(x\) in the function \(f(x)\). So: \(f(g(h(x))) = f(4x^{2})= 4x^{2}-1.\) There are no constraints, meaning that the composition is defined for all real numbers. Hence, the domain of \(f(g(h(x)))\) is \(-\infty< x <+\infty.\)

Key Concepts

Domain of a Function

Understanding the domain of a function is critical as it tells us where the function is applicable. The domain of a function includes all possible input values (usually represented by 'x') that allow the function to work without errors.

For example, consider a mathematical function like \(f(x) = \frac{1}{x}\). The domain would exclude \(x = 0\) because dividing by zero is undefined. Hence, the domain is all real numbers except zero.

In the composition of functions like \(f \circ g \circ h\), the domain is influenced by each function in the composition chain. You must ensure that each function's domain is compatible with the output from the previous function. Thus, the domain of the overall composition becomes the intersection of the domains where every intermediate output acts as a valid input for the next function.

Real Numbers

Real numbers are the cornerstone of many mathematical functions. They include every number along the number line, encompassing integers, rational numbers, and irrational numbers. Real numbers are infinitely continuous, covering every possible point between any two numbers, no matter how close they are.

Here's what real numbers encompass:

• Integers: Whole numbers, both positive and negative, including zero (e.g., -3, 0, 7).
• Rational Numbers: Numbers that can be expressed as a fraction of two integers, such as \(\frac{1}{2}\).
• Irrational Numbers: Numbers that cannot be expressed as simple fractions, such as \(\sqrt{2}\) and \(\pi\).

Every value in the domain of our composition \(f \circ g \circ h\) is a real number, meaning it includes all values between \(-\infty\) and \(+\infty\). These numbers provide the full range for inputs and outputs for functions without restriction, unless otherwise specified by the context of the function.

Mathematical Functions

Mathematical functions are rules that relate an input to a specific output. Every function follows its own unique formula, determining how it interprets each input value that falls within its domain. Functions can come in various forms, such as linear, quadratic, exponential, and more.

Let's explore the types of functions encountered in our exercise:

• Linear Functions: They have the simplest form, expressed as \(f(x) = mx + b\), where the graph forms a straight line.
• Quadratic Functions: Shaped like a parabola, they follow the general form \(f(x) = ax^2 + bx + c\).

In the given task, we dealt with functions \(f(x) = x-1\), \(g(x)=4x\), and \(h(x)=x^2\), each representing different types of mathematical functions.

When composing functions such as \(f \circ g \circ h\), you substitute the outputs from one function as the input for the next, building a sequence of transformations. This composition allows for complex computation but requires understanding how each function operates individually.