From rational functions to inverse functions, meet four brand-new graph shapes
These are the functions you learn right after linear and quadratic functions. Rational and irrational functions show new graph shapes, while composite and inverse functions show how to connect functions together and flip them around.
Rational function y=k/(xāp)+q ā a two-branch curve that gets closer and closer to two lines (asymptotes) but never touches them.
Irrational function y=Ā±ā(Ā±(xāp))+q ā the inside of the root (ā) can never be negative, so the domain is restricted to one side.
Composite function (fāg)(x)=f(g(x)) ā you feed the output of one function into another function as its input. Swap the order and the result usually changes too.
Inverse function fā»Ā¹(x) ā a function with its input and output completely swapped. Its graph is always symmetric about the line y=x.
y = k/(xāp) + q. Since x=p makes the denominator 0, it's undefined there, so the graph only ever gets infinitely close to the two lines (asymptotes) x=p and y=q without ever touching them.
p (x-coordinate of the asymptote)1
q (y-coordinate of the asymptote)-1
|k|2
y = Ā±ā(Ā±(xāp)) + q. The value inside the root (ā) can never be less than 0, so the domain is restricted to only one direction from p. The gray-shaded region is where the graph can't exist.
p (x-coordinate of the starting point)0
q (y-coordinate of the starting point)0
The composite function (fāg)(x)=f(g(x)) feeds x into g first, then feeds that result into f. Try changing the pair of functions and the input value to compare (fāg)(x) and (gāf)(x).
Input x1
The inverse of the linear function f(x)=ax+b is fā»Ā¹(x)=(xāb)/a. Reflecting the point (t, f(t)) across the line y=x always gives the point (f(t), t) on fā»Ā¹ ā that's why the graphs of f and fā»Ā¹ are symmetric about y=x.