Problem 22 - Algebra 2 and Trigonometry

Question

If \(\mathrm{g}=\\{(x, y) : y=7-x\\},\) find \(\mathrm{g}^{-1}\) if it exists. Is it possible for a function to be its own inverse?

Step-by-Step Solution

Answer

The inverse is the same function, \( g^{-1}(x) = 7 - x \). Yes, a function can be its own inverse.

Step 1: Understanding the Inverse Function

To find the inverse of a function, we need to swap the roles of the domain and the range. This means that instead of having an equation where y is expressed in terms of x, we need a new function where x is expressed in terms of y. For the function \( y = 7 - x \), we need to solve for x in terms of y.

Step 2: Solving for x in Terms of y

Start by expressing x in terms of y from the equation \( y = 7 - x \). Rearrange this equation to solve for x. Add x to both sides to get \( y + x = 7 \), then subtract y from both sides to obtain \( x = 7 - y \).

Step 3: Finding the Inverse Function

Now that we have expressed x in terms of y (\( x = 7 - y \)), swap x and y to find the inverse function. Replace x with y and y with x, resulting in the inverse function \( y = 7 - x \).

Step 4: Check If the Function is Its Own Inverse

Notice that the original function \( y = 7 - x \) and the inverse function \( y = 7 - x \) are identical. This confirms that the function is indeed its own inverse.

Key Concepts

Function Inversion

Function inversion is a process where we reverse the roles of the input and output in a function. When we talk about a function, we typically have a relationship between two sets of numbers or variables, where every input from the first set is assigned to exactly one output in the second set. Inverting a function involves swapping these assignments, making the output act as the new input and vice versa.

• Start with a function in the form of an equation, like the function given in the exercise, \( y = 7 - x \).
• The goal is to express the original input variable \( x \) in terms of the output variable \( y \).
• By rearranging the original equation, you solve for \( x \) as \( x = 7 - y \).
• Finally, you swap the variables again to write the inverse function, resulting in \( y = 7 - x \).

The concept behind this is that every pair \((x, y)\) in the original function becomes \((y, x)\) in the inverse function. It's like a reflection along the line \( y = x \). Understanding this basic principle helps in tackling more complex inverse function problems.

Domain and Range

In any function, domain and range play vital roles. The domain is the set of all possible input values (\( x \)-values) for which the function is defined. Conversely, the range is the set of all possible output values (\( y \)-values) the function can produce. When we find the inverse of a function, these sets swap their roles.In our function \( y = 7 - x \):

• The domain is all real numbers, since \( x \) can be any value.
• The range is also all real numbers because \( y \) takes on every value as \( x \) varies.

For the inverse function, the original range becomes the new domain, and the original domain becomes the new range. This swapping impacts how we think about these sets when evaluating or plotting inverse functions. Moreover, both domain and range must cover all real numbers for any meaningful function inversion.

Self-Inverse Functions

A function is termed 'self-inverse' if it acts as its own inverse. This means applying the function twice returns you to your original input value. In mathematical terms, if the function \( f(x) \) satisfies \( f(f(x)) = x \) for all inputs \( x \) in its domain, it is self-inverse.In the exercise, the function \( y = 7 - x \) turns out to be its own inverse because its inverse is identical to the original function.

• Such functions are neat and elegant in mathematics.
• Common examples of self-inverse functions are \( f(x) = -x \) and \( f(x) = \frac{1}{x} \) where these functions map an input to itself after two applications.

Identifying self-inverse functions can save a lot of effort, as solving for their inverse is unnecessary beyond the initial check. Recognizing patterns that suggest a function may be self-inverse helps simplify problems quickly.