Problem 8 - 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. $$ f(x)=\frac{2}{x-5} \text { and } g(x)=\frac{2}{x}+5 $$

Step-by-Step Solution

Answer

The functions \(f(x)=\frac{2}{x-5}\) and \(g(x)=\frac{2}{x}+5\) are inverses of each other.

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

To start, substitute \(g(x)\) into \(f(x)\). This means wherever there is an 'x' in \(f(x)\), it should be replaced with \(\frac{2}{x}+5\). So, \(f(g(x))\) becomes \(f(\frac{2}{x}+5) = \frac{2}{(\frac{2}{x}+5) - 5}\). Simplify this expression.

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

Simplify the expression \(f(\frac{2}{x}+5) = \frac{2}{(\frac{2}{x}+5) - 5}\) by utilizing the property of fractions, that is, the determinant of fractions is equal to the determinant of their numerators and denominators; here \(\frac{2}{(\frac{2}{x}+5)-5} = \frac{2}{\frac{2+5x}{x}-5} = \frac{2x}{2+5x-5x} = \frac{2x}{2} = x\).

Step 3: Find \(g(f(x))\)

Now, do the same process with \(g(f(x))\). Substitute \(f(x)\) into \(g(x)\) resulting in \(g(f(x)) = g(\frac{2}{(x-5)}) = \frac{2}{\frac{2}{(x-5)}} + 5\). Simplify this expression.

Step 4: Simplify \(g(f(x))\)

Simplify the expression \(g(\frac{2}{(x-5)}) = \frac{2}{\frac{2}{(x-5)}} + 5\) to \( g(\frac{2}{(x-5)}) = x-5 + 5 = x\).

Step 5: Check Inverses

If two functions are inverses of each other, then both \(f(g(x))\) and \(g(f(x))\) should equal to x. In this case, \(f(g(x)) = x\) and \(g(f(x)) = x\), so we can conclude that the two given functions are indeed inverses of each other.

Key Concepts

Understanding Function Inverses

In the realm of mathematics, the concept of function inverses is akin to finding a mirroring relationship between two functions. When you have two functions, say, f and g, their inverses effectively reverse each other’s processes. In mathematical terms, if f(g(x)) = x and g(f(x)) = x, the functions f and g are inverses.

Imagine a scenario where one function, perhaps f, is responsible for packing lunch into a box, and g, the inverse function, perfectly unpacks the box to reveal the original lunch. If each function perfectly reverses the action of the other, leaving us with what we started, they are true inverses.

This reciprocal relationship is not only elegant but also immensely useful in solving equations and understanding transformations within coordinate systems.

Algebraic Manipulation Masterclass

Algebraic manipulation is the art of reworking equations and expressions into a form that is more suitable for solving a particular problem. It’s akin to rearranging the pieces of a puzzle to complete the picture. To master this skill, one must be comfortable with various operations, such as distributing, factoring, and combining like terms.

The process requires a clear understanding of algebraic properties, including the distributive property, associative property, and commutative property. Through these techniques, complex expressions are rendered into simpler, more digestible forms, allowing us to unveil the inherent relationships within mathematical equations.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics, which helps in reducing complexity and uncovering the core elements of an algebraic expression. This process involves eliminating parentheses, combining like terms, and reducing fractions to their simplest form.

Consider the following: if you have a basket filled with different fruits separated by type into smaller baskets, simplifying an expression would be like combining all the like fruits together, making it easier to count and manage. In the context of the exercise provided, simplifying allows us to reduce complex fractions and distill the expression down to its simplest form, aiding in the revelation that the two functions are indeed inverses.

Properties of Fractions Explored

Fractions represent a part of a whole, and understanding their properties is essential for effective mathematical communication. This involves comprehending how to combine, reduce, and compare fractional quantities. Key properties include the ability to find common denominators, cross-multiply, and simplify complex fraction equations.

In the context of our functions, applying these properties allows us to manipulate and simplify expressions involving fractions. By doing so, we deftly untangle the convolutions of fractional operations, enabling us to confirm the inverse relationship between two functions. With care and practice, mastery of fractions becomes a powerful tool in the mathematician’s toolkit.