Problem 38 - Algebra and Trigonometry

Question

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\frac{1}{\sqrt{x}}, \quad g(x)=x^{2}-4 x $$

Step-by-Step Solution

Answer

Domain for \(f \circ g\): \((4, \infty)\); for \(g \circ f\): \((0, \infty)\); for \(f \circ f\): \((0, \infty)\); for \(g \circ g\): \((-\infty, \infty)\).

Step 1: Find \(f \circ g\) and Its Domain

The composition \(f \circ g\) involves plugging \(g(x) = x^2 - 4x\) into \(f(x) = \frac{1}{\sqrt{x}}\). Thus,\[f \circ g(x) = f(g(x)) = f(x^2 - 4x) = \frac{1}{\sqrt{x^2 - 4x}}.\]For the domain, we need \(x^2 - 4x > 0\) and \(x eq 0\) so:1. Solve \(x^2 - 4x > 0\): - Factor: \(x(x - 4) > 0\) - Solve using test intervals from critical points \(x = 0\) and \(x = 4\).2. The solution gives intervals \((-\infty, 0) \cup (4, \infty)\) as valid for \(f \circ g\), excluding points making the expression under the square root zero.

Step 2: Find \(g \circ f\) and Its Domain

The composition \(g \circ f\) involves plugging \(f(x) = \frac{1}{\sqrt{x}}\) into \(g(x) = x^2 - 4x\). Thus,\[g \circ f(x) = g(f(x)) = g\left(\frac{1}{\sqrt{x}}\right) = \left(\frac{1}{\sqrt{x}}\right)^2 - 4\left(\frac{1}{\sqrt{x}}\right) = \frac{1}{x} - \frac{4}{\sqrt{x}}.\]For the domain of \(g \circ f(x)\), \(x > 0\) as the square root in \(f(x)\) requires \(x > 0\). There are no further restrictions.

Step 3: Find \(f \circ f\) and Its Domain

For \(f \circ f(x)\), compose \(f(x)\) with itself:\[f \circ f(x) = f(f(x)) = f\left(\frac{1}{\sqrt{x}}\right) = \frac{1}{\sqrt{\frac{1}{\sqrt{x}}}} = \sqrt[4]{x}.\]The domain of \(f \circ f(x)\) is limited by the requirement of \(f(x)\), meaning \(x > 0\) to ensure the expression under the square root is defined, resulting in \(x > 0\).

Step 4: Find \(g \circ g\) and Its Domain

For \(g \circ g(x)\), compose \(g(x)\) with itself:\[g \circ g(x) = g(g(x)) = g(x^2 - 4x) = (x^2 - 4x)^2 - 4(x^2 - 4x),\]which simplifies to \(x^4 - 8x^3 + 20x^2 - 16x\).There are no domain restrictions outside of those naturally imposed by polynomial functions like \(g \circ g(x)\). Hence, the domain is all real numbers, \((-\infty, \infty)\).

Key Concepts

Understanding Function Domains

The domain of a function is simply the set of all possible inputs (typically "x" values) that the function can accept without causing any mathematical errors. Ensuring a function's domain involves checking for values that don't work, like dividing by zero or taking the square root of a negative number.
For example, if we consider the function \( f(x) = \frac{1}{\sqrt{x}} \), the expression under the square root must be positive, meaning \( x > 0 \). If \( x \) were zero or negative, \( f(x) \) would be undefined.
In composite functions like \( f \circ g \), determining the domain involves verifying that the output of the function \( g(x) \) creates a valid input for \( f(x) \). Overall, domains are crucial for understanding where a function behaves well.

Exploring Function Composition

Function composition is all about creating a new function by applying one function to the results of another. It's like following a two-step recipe, where each function does part of the work. Think of it as putting one function into another, not unlike having two machines where the output of the first is the input for the second.
Given two functions, \( f(x) = \frac{1}{\sqrt{x}} \) and \( g(x) = x^2 - 4x \), the composition \( f \circ g \) is formed by substituting \( g(x) \) into \( f(x) \), leading to \( f \circ g(x) = \frac{1}{\sqrt{x^2 - 4x}} \). Here, we ensure \( x^2 - 4x > 0 \) for the composition to be defined.
Similarly, for \( g \circ f \), we switch the order and substitute \( f(x) \) into \( g(x) \), resulting in \( g \circ f(x) = \frac{1}{x} - \frac{4}{\sqrt{x}} \). Going through compositions ensures that we discover how functions interact with each other.

Insights into Polynomial Functions

Polynomial functions are expressions that combine variables, coefficients, and non-negative integer exponents. These are fundamental to algebra, forming smooth, continuous curves that extend infinitely unless restricted by a domain.
For instance, \( g(x) = x^2 - 4x \) represents a simple polynomial where the domain is all real numbers. When polynomials are composed, like in \( g \circ g \), we form higher-degree polynomials, such as \( x^4 - 8x^3 + 20x^2 - 16x \).
The beauty of polynomial functions is their predictability: they are defined for every real number unless interactions with other functions add restrictions. Compositions with polynomials often lead to these higher-degree functions, which still maintain their inherent predictability and smoothness.