Problem 43 - Algebra for College Students

Question

If \(f(x)=x^{2}\) and \(g(x)=\sqrt{x}\), with both having a domain of the set of nonnegative real numbers, then show that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\).

Step-by-Step Solution

Answer

\((f \circ g)(x) = x\) and \((g \circ f)(x) = x\) for all nonnegative real \(x\).

Step 1: Understand the Composition of Functions

To solve the problem, we need to understand what function composition means. The notation \((f \circ g)(x)\) means applying \(g(x)\) first and then applying \(f\) to the result of \(g(x)\). Similarly, \((g \circ f)(x)\) means applying \(f(x)\) first and then applying \(g\) to the result of \(f(x)\).

Step 2: Compute \((f \circ g)(x)\)

The function \(g(x) = \sqrt{x}\) is applied first. Then, the output of \(g(x)\) becomes the input for the function \(f(x) = x^2\). Thus, we have:\[ (f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^2. \]Since \((\sqrt{x})^2 = x\), it follows that \((f \circ g)(x) = x\).

Step 3: Compute \((g \circ f)(x)\)

The function \(f(x) = x^2\) is applied first. Then, the output of \(f(x)\) becomes the input for the function \(g(x) = \sqrt{x}\). Thus, we have:\[ (g \circ f)(x) = g(f(x)) = g(x^2) = \sqrt{x^2}. \]Since \(\sqrt{x^2} = x\) for nonnegative \(x\), it follows that \((g \circ f)(x) = x\).

Step 4: Conclude the Result

In both cases, we have shown that the composition of the functions returns the original variable \(x\). Thus, \((f \circ g)(x) = x\) and \((g \circ f)(x) = x\) for all nonnegative real values of \(x\).

Key Concepts

Inverse Functions

Inverse functions are special pairs of functions where each undoes the action of the other, meaning when you apply one function and then the other, you end up back at your starting point.
For example, if you have a function \( f(x) \) that squares its input \( f(x) = x^2 \), and another function \( g(x) \) that takes the square root \( g(x) = \sqrt{x} \), they work as inverse functions over the domain of nonnegative real numbers.
This is because applying \( g \) after \( f \), or \( f \) after \( g \), always brings back the original value \( x \) from nonnegative values. In simpler terms, the two functions "undo" each other's operations when composed:

• \((f \circ g)(x) = x\)
• \((g \circ f)(x) = x\)

Remember, an inverse function essentially "reverses" another function's effect. This pairing only holds true when dealing with valid domains that ensure the operations are well-defined and produce real numbers.

Domain and Range

Understanding the domain and range of a function is crucial when working with compositions, and it helps ensure that the compositions are valid. The domain is the set of all possible input values that a function can accept, while the range is the set of all potential outputs the function can produce.
For both \( f(x) = x^2 \) and \( g(x) = \sqrt{x} \), the domain is restricted to nonnegative real numbers. This restriction ensures that every output from \( g(x) \) is a valid input for \( f(x) \), and vice versa.
The range of \( f(x) \) is also nonnegative real numbers since squaring any real number cannot result in a negative. The function \( g(x) \), limited to nonnegative inputs, also results in nonnegative outputs.

• The domain of both functions is \([0, \infty)\)
• The range of \(f(x)\) is \([0, \infty)\)
• The range of \(g(x)\) is \([0, \infty)\)

Keeping these considerations in check ensures that the function compositions we perform are valid and that the end result is meaningful.

Algebraic Functions

Algebraic functions are constructed from polynomials and involve operations like addition, subtraction, multiplication, division, and taking roots. The functions \(f(x) = x^2\) and \(g(x) = \sqrt{x}\) are classic examples.
These algebraic functions are well-defined and allow for various manipulations, making them an essential part of functional analysis and composition. Squaring is an operation that naturally follows algebraic principles as it involves multiplying a term by itself.
Similarly, taking the square root is an algebraic operation that seeks to find a value which, when squared, returns the original number.

• \(f(x) = x^2\) is a polynomial function of degree 2.
• \(g(x) = \sqrt{x}\) can be described as \(x^{1/2}\), involving the operation of taking a square root.

Both functions, in the context of nonnegative real numbers, continuously map these numbers to equally acceptable algebraic values, forming an inverse pair as shown in the function compositions. They illustrate basic but vital algebraic operations that are frequently encountered in mathematical processes and real-world problem solving.