11th Grade Q1

Creating a table of values for an inverse function involves reversing the roles of the input and output values for the original function.

Method

Let's assume a function $f(x)$ is invertible. How do we create the table of values for it's inverse?

1. Solve for the inverse function: If you have the original function, calculate it's inverse.
2. Pick values for the original function: Create a table of input values ($x$) and compute their corresponding output values ($y$) using the original function.
3. Switch the roles for the inverse: Flip the input ($x$) and output ($y$) values from the original function to get values for the inverse function.
4. Verify the inverse relationship: If you have the original function, check that the $x$ and $y$ values in the table for the inverse satisfy the original function when switched.

Example

Let’s create a table for the inverse $f(x)=2x+3$.

1: Solve for the inverse function

To find the inverse, you solve $y=2x+3$ for $x$.

The inverse is function $f^{-1}(x)$ is the right hand side with $y$ replaced by $x$,

2: Pick values for the original function

Let's consider $x$-values of $0,1,2$ and $3$. The table for the original function, $y=f(x)=2x+3$ is

3: Switch the roles for the inverse

Switch $x$ and $y$ values from the original table. The table for the inverse function $y=f^{-1}(x)=\frac{1}{2}(x-3)$ is

4: Verify the inverse relationship

Let's verify the table by substituting $x$ values into the inverse function $f^{-1}(x)=\frac{1}{2}(x-3)$.

• For $x=3$ in the inverse table: $f^{-1}(3)=\frac{1}{2}(3-3)=0$.
• For $x=5$ in the inverse table: $f^{-1}(5)=\frac{1}{2}(5-3)=1$.

Both values match the original table when the roles are reversed. This confirms the inverse relationship.

Now you have a table of values for both the original function and its inverse!