Interpreting a linear function and its inverse steps

Key concept: This item shows a simple linear function rewritten through inverse-style steps, so the goal is to recognize the algebraic rule being represented.

Step by step

Recognize the function

The displayed work centers on

$F(x)=2-x.$

This is a linear function with slope $-1$ and y-intercept $2$. The rewritten steps

$y=2-x$

then

$x=2-y$

show the relationship being rearranged by solving for the other variable.

What the algebra means

Starting from

$y=2-x,$

add $x$ to both sides:

$x+y=2.$

Then subtract $y$ or subtract $x$ depending on the variable you want to isolate. These steps do not change the function; they simply rewrite the same equation in different forms.

Graph interpretation

The graph of $y=2-x$ is a straight line that crosses the y-axis at $(0,2)$ and the x-axis at $(2,0)$. Because the slope is $-1$, the line decreases at a constant rate.

If the task is to identify the inverse, note that this particular function is its own inverse:

$f(x)=2-x \quad \Rightarrow \quad f^{-1}(x)=2-x.$

You can verify this by swapping $x$ and $y$ and solving:

$x=2-y \Rightarrow y=2-x.$

So the same expression appears again.

Final conclusion

The function shown is a decreasing linear function with equation

$\boxed{F(x)=2-x}.$

It is also self-inverse, which is why the algebraic rearrangement returns the same rule.

Pitfall alert

A frequent mistake is to think that $x=2-y$ is a new function instead of the same relationship rewritten. It is not a different rule; it is just the original equation solved for $x$. Another issue is mixing up the inverse with the reciprocal. The inverse of $2-x$ is not $1/(2-x)$; it is the same linear expression after swapping variables. Students also sometimes forget that a slope of $-1$ means the line goes down one unit for every one unit to the right, which helps when sketching or checking the equation.

Try different conditions

If the function were $F(x)=5-x$, the same process would still work: write $y=5-x$, swap variables to get $x=5-y$, and then solve for $y$ to confirm that the inverse is again $5-x$. If the function were $F(x)=2+x$ instead, the inverse would also be itself after swapping and solving, because lines of the form $y=x+b$ are self-inverse only when the transformation is symmetric in the right way. Changing the constant shifts the intercept but keeps the same algebraic procedure.

Further reading

linear function, inverse function, slope-intercept form