Problem 9 - College Algebra

Question

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$ f(x)=-x \text { and } g(x)=-x $$

Step-by-Step Solution

Answer

The functions \(f(x) = -x\) and \(g(x) = -x\) are inverses of each other as \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Find \(f(g(x))\)

First replace \(x\) in \(f(x) = -x\) with \(g(x)\). Since \(g(x) = -x\), \(f(g(x))\) equals \(-(-x)\), which simplifies to \(x\).

Step 2: Find \(g(f(x))\)

Now replace \(x\) in \(g(x) = -x\) with \(f(x)\). Because \(f(x) = -x\), \(g(f(x))\) equals \(-(-x)\), which simplifies to \(x\).

Step 3: Check if \(f\) and \(g\) are inverses

The results for both \(f(g(x))\) and \(g(f(x))\) are \(x\). Moreover, substituting \(f(x)\) into \(x = g(x)\) gives \(x = -(-x)\), which simplifies to \(x = x\). This shows every element of the range of one function corresponds to exactly one element of the domain of the other function. Thus \(f\) and \(g\) are inverses of each other.

Key Concepts

Composition of Functions

The composition of functions is a mathematical operation in which two functions are combined into a single function. This is done by applying one function to the result of another function. For instance, given two functions, say, \(f(x)\) and \(g(x)\), the composition \(f(g(x))\) means you first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\).

A crucial point about composition is that the domain of the composite function will be affected by both the domain of the first function applied and the range of the second function. It is also important to note that the order in which functions are composed matters – in general, \(f(g(x))\) is not the same as \(g(f(x))\), highlighting the non-commutative nature of function composition.

In the provided exercise, since \(f(x) = -x\) and \(g(x) = -x\), both compositions, \(f(g(x))\) and \(g(f(x))\), result in the original input \(x\), evidencing a special case where the composition of functions is commutative.

Algebraic Operations

Algebraic operations include basic arithmetic processes like addition, subtraction, multiplication, and division, which can be utilized to manipulate algebraic expressions and functions. When performing algebraic operations on functions, you replace variables with function expressions as required.

For instance, in algebraic operations involving functions, if you want to subtract function \(f(x)\) from function \(g(x)\), you would compute \(g(x) - f(x)\), which requires deductive manipulation of the expressions for \(f(x)\) and \(g(x)\). Similarly, when finding the composition of functions, which is an algebraic operation, you're essentially substituting one function into another and simplifying.

In the given exercise, we applied algebraic operations to the functions \(f(x)\) and \(g(x)\) to find their composition. By substituting \(-x\) for \(x\) and simplifying, we performed algebraic operations that led us to the conclusion that \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), a crucial insight for understanding the relationship between these functions.

Function Inverses

A function inverse is a special type of function that reverses the action of another function. For a function \(f\) with an inverse \(f^{-1}\), the relationship is characterized by the equations \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\), meaning applying the function and its inverse in succession results in the original value.

For a function to have an inverse, it must be bijective; that is, it needs to be both injective (one-to-one) and surjective (onto). Practically, this implies that every output is the result of one and only one input, ensuring that the inverse function can 'trace back' each output to a unique input.

In the context of the exercise, by illustrating that both \(f(g(x))\) and \(g(f(x))\) simplify to \(x\), we prove that \(f(x)\) and \(g(x)\) are inverses. This is because, for both functions, applying one and then the other brings us back to our starting point, thereby satisfying the defining condition of inverse functions.