Identifying Inverse Functions: Criteria and Assessment

How can you tell if functions are inverse functions?

To determine if two functions are inverse functions, you can use the following criteria and assessments:

1. Composition of Functions:

   • Given two functions, $f(x)$ and $g(x)$, check if the composition $f(g(x))$ is equal to $x$ for all $x$ in the domain of $g$, and if $g(f(x))$ is equal to $x$ for all $x$ in the domain of $f$.

2. Symbolic Representation:

   • If $f(g(x)) = x$ and $g(f(x)) = x$, then $g(x)$ is the inverse function of $f(x$, and vice versa.

3. Algebraic Verification:

   • For functions $f$ and $g$ to be inverses, their composition should satisfy the following equations:
     • $f(g(x)) = x$ (for all $x$ in the domain of $g$)
     • $g(f(x)) = x$ (for all $x$ in the domain of $f$)

4. One-to-One Property:

   • A function must be one-to-one (injective) for it to have an inverse. This means that each distinct input corresponds to a unique output.

5. Domain and Range:

   • Ensure that the domain and range of the functions are chosen such that the inverse exists. In some cases, you may need to restrict the domain to make the functions one-to-one.

6. Graphical Representation:

   • On a coordinate plane, the graphs of inverse functions are reflections of each other across the line $y = x$.

Example:

Let $f(x) = 2x + 3$ and $g(x) = \frac{1}{2}(x - 3)$ be two functions.

1. Composition:

   • Calculate $f(g(x))$ and $g(f(x))$.
     • $f(g(x)) = 2\left(\frac{1}{2}(x - 3)\right) + 3 = x$
     • $g(f(x)) = \frac{1}{2}(2x + 3 - 3) = x$

2. Verification:

   • Since $f(g(x)) = x$ and $g(f(x)) = x$, $g(x)$ is the inverse function of $f(x)$, and vice versa.

In this example, $f(x) = 2x + 3$ and $g(x) = \frac{1}{2}(x - 3)$ are indeed inverse functions.

By applying these criteria and assessments, you can determine whether two functions are inverses of each other.

Indications of functions being inverse functions?