Question 432·Hard·Algebra

No Desmos needed. The swapped values give an exact slope by hand. In $g(a)=b$, $a$ goes in and $b$ comes out, so $(a,b)$ is on the line. Likewise, $g(b)=a$ gives $(b,a)$. The input comes first in each point.

The slope is the change in output divided by the change in input. Subtract the coordinates of $(a,b)$ from $(b,a)$ in the same order:

The numerator is the opposite of the denominator. Rewrite it:

Because $a$ and $b$ are distinct, $b-a$ isn’t zero. Cancel it:

A line with slope $-1$ has the form $g(x)=-x+c$, where the intercept $c$ is its output at input $0$. Put in the point $(a,b)$; $b$ is not automatically the intercept:

Add $a$ to both sides to find the intercept:

Replace $c$ with $a+b$ in the line: $g(x)=-x+a+b$. This rule has the slope of the two given points and passes through $(a,b)$, so it is the required linear function. Choice B.