Question

Do functions of the form \(y=x^{m / n}\), where \(m\) and \(n\) are positive integers, have inverse functions? Justify your answer with examples.

Short answer

Yes, functions of the form \(y=x^{m/n}\) where \(m\) and \(n\) are positive integers do have inverse functions. Their inverses are of the form \(y=x^{n/m}\), and an example would be the function \(y=x^{2/1}\) with inverse \(y=x^{1/2}\).

Step-by-step solution

Understand what an inverse function is

An inverse function is a function that 'undoes' the work of another function. It is formally defined as a function \(g(x)\) being the inverse of a function \(f(x)\) if for every \(x\) in the domain of \(f\), \(f(g(x)) = x\) and for every \(x\) in the domain of \(g\), \(g(f(x)) = x\).

Prove that functions of the form \(y=x^{m / n}\) where \(m\) and \(n\) are positive integers have inverse function

Functions of the form \(y = x^{m/n}\) will have inverse functions provided that they pass the horizontal line test, which states that any horizontal line drawn through the function will intersect the function at most once. The inverse function of \(y = x^{m/n}\) can be found by switching \(x\) and \(y\), hence \(x = y^{m/n}\). By solving this equation, you get \(y = x^{n/m}\), which is a valid function given the conditions. Therefore, functions of the form \(y=x^{m / n}\) where \(m\) and \(n\) are positive integers do have inverse functions.

Justify with examples

Take \(y=x^{2/1}\) as an example. The function passes the horizontal line test for positive \(x\), and therefore it has an inverse inside its domain. Its inverse is \(y=x^{1/2}\) (or in other words, the square root of \(x\)).

Key concepts

Horizontal Line Test

The horizontal line test is a method to determine if a function has an inverse. To apply this test, visualize or graph a function and imagine drawing a horizontal line across it.

The function passes the horizontal line test if a horizontal line intersects the function at most once at any given y-value. This indicates that the function is one-to-one (injective) and thus has an inverse.

• Functions that pass this test have unique outputs for each input, meaning they never repeat a y-value over their domain without repeating the x-value.
• For example, the function \(y = x\), being a straight diagonal line through the origin, passes this test as any horizontal line will cross the graph just once.

For more complex functions like those of the form \(y = x^{m/n}\), you need to consider their behavior over specific domains. If the power function's graph passes the test over its relevant domain (like parts where \(x \geq 0\)), it has an inverse within that domain.

Rational Exponent Functions

Rational exponent functions are mathematical expressions where the variable is raised to a rational power, i.e., a fraction. These functions, such as \(y = x^{m/n}\), show interesting behaviors and have specific properties:

• The base, \(x\), is raised to a power given by \(m/n\), where \(m\) and \(n\) are positive integers.
• This can be rewritten as \( (x^m)^{1/n} \), showcasing the duality of these functions as both power and root functions.
• Specifically, \(m\) represents the power, while \(n\) indicates the root degree.

Understanding rational exponent functions helps in identifying their inverses. For example, \(y = x^{2/3}\) is equivalent to the cube root of \(x^2\). The inverse would involve switching input and output, reflecting the variable across the line \(y = x\), and solving algebraically.

Inverse of Power Functions

Inverse functions play a crucial role in understanding the capabilities and constraints of mathematical operations. For functions with the form \(y = x^{m/n}\), determining if an inverse exists is about flipping the roles of x and y and ensuring the function remains valid.

The inverse of these power functions can be derived by first expressing the given function in terms of its inverse.

• Start by swapping \(x\) and \(y\) in the equation: \(x = y^{m/n}\).
• The next step is solving for \(y\), which involves raising both sides to the reciprocal power \(n/m\). This results in \(y = x^{n/m}\), the inverse function.

Remember, the existence of an inverse is conditional on the original function passing the horizontal line test, indicating a one-to-one relationship over its domain. By confirming the function meets these criteria, we ensure its inverse persists without violating fundamental algebraic properties.