Problem 64 - Calculus Single Variable

Question

Find a function \(g\) such that \(g \circ f=h\) \(f(x)=(x-1) / x^{2}, h(x)=x^{2} /(x-1)\)

Step-by-Step Solution

Answer

The function \( g(x) = \frac{1}{x} \) satisfies \( g \circ f = h \).

Step 1: Understand the Composition

We need to find a function \( g(x) \) such that when \( f(x) = \frac{x-1}{x^2} \) is composed with \( g \), it equals \( h(x) = \frac{x^2}{x-1} \). In mathematical terms, \( g(f(x)) = h(x) \).

Step 2: Set the Composition Equation

Express the equation \( g(f(x)) = h(x) \) by substituting \( f(x) \) into \( g \): \[ g \left( \frac{x-1}{x^2} \right) = \frac{x^2}{x-1} \].

Step 3: Simplify the Input of Function g

Let \( y = \frac{x-1}{x^2} \). Therefore, we need to find \( g(y) \) such that \( g(y) = \frac{x^2}{x-1} \) when \( y = \frac{x-1}{x^2} \).

Step 4: Solve for x in Terms of y

From \( y = \frac{x-1}{x^2} \), express \( x \) in terms of \( y \): Solving the equation gives \( yx^2 = x - 1 \), leading to \( x^2 = \frac{x-1}{y} \).

Step 5: Find g(y) with the Inverse Expression

Since \( y x^2 = x - 1 \), to find \( g(y) \), solve for \( x \): rearrange to get \( x = \frac{1}{1-xy} \), meaning \( x = \frac{1}{1-rac{y}{x}} \). Substitute \( y = \frac{x-1}{x^2} \) back. Thus, \( g(y) = \frac{1}{y} \).

Step 6: Verify g(y)

Verify that \( g(y) = \frac{1}{y} \) satisfies the equation \( g(y) = \frac{x^2}{x-1} \). It does indeed work when substituting back: \( g \left( \frac{x-1}{x^2} \right) = \frac{x^2}{x-1} \).

Key Concepts

Inverse Functions

Inverse functions are fascinating tools in mathematics. They essentially "undo" what a given function does. Imagine you have a specific function, such as \( f(x) \), which transforms an input \( x \) into an output \( f(x) \). An inverse function, typically denoted as \( f^{-1}(x) \), will take that output and return it back to the original input. This is why they are called 'inverses'—they reverse the operation of the original function.

For a function to have an inverse, it must be one-to-one (bijective), meaning every output is paired with exactly one input. If this isn't the case, the function can't have an inverse because the reversal wouldn't be unique.

If you want to find the inverse of a function, you typically follow these steps:

• Switch the roles of \( y \) and \( x \) in the function's equation.
• Solve for \( y \) in terms of \( x \).
• Replace \( y \) with \( f^{-1}(x) \) to denote the inverse function.

Inverse functions have applications in various areas of mathematics and are especially useful in solving equations and understanding function compositions.

Solving Equations

Solving equations is a fundamental skill in mathematics. It's about finding the value of the variable that makes the equation true. When it comes to more complex equations, such as those including fractions or multiple variables, it can be vastly helpful to isolate the variable step by step.

To solve an equation like \( y x^2 = x - 1 \), where you are looking to express \( x \) in terms of \( y \), you should:

• Simplify the equation as much as possible initially, possibly by moving all terms involving the same variable to one side.
• If fractions are involved, eliminate them by multiplying every term by the least common denominator.
• Reorganize and factor if needed to clearly isolate the variable of interest.

Once \( x \) is expressed in terms of \( y \), it becomes possible to understand how changes in \( y \) affect \( x \), offering deeper insights into the relationship defined by the original equation.

Mathematical Expressions

Mathematical expressions are combinations of variables, numbers, and operation symbols standing together to represent a certain value or relationship. They are the building blocks of equations.

Understanding how to manipulate and simplify these expressions is crucial when dealing with complex mathematical problems. Here are some key aspects:

• Terms in the expression are separated by plus or minus signs. Each term is a part of the expression that can be a coefficient, variable, or both.
• Understand how exponents and roots work in multiplication and division. Knowing these rules help in simplifying or transforming expressions.
• Identifying like terms is important: you can only combine terms that have the exact same variables raised to the same power.

When you're given an expression like \( g \( \frac{x-1}{x^2} \) \), part of your task is to simplify or manipulate this until you derive the correct form required by the problem, such as finding \( g(y) \) in the original exercise.