Once you've learned linear and quadratic functions, High School 1 introduces four brand-new functions. Rational and irrational functions show you a new kind of graph shape, while composite and inverse functions show you how to build a new function using a function itself as raw material.
Let's start with rational functions. An expression of the form y = k/(x−p) + q is called a rational function. Here, x=p makes the denominator 0, so it can never be defined at that point. That's why the graph splits into two branches that only ever get infinitely close to the vertical line x=p and the horizontal line y=q, without ever touching them. A line that a graph only approaches but never meets like this is called an asymptote. It's much easier to understand if you think of y=k/(x−p)+q as the simplest form, y=k/x (an inverse proportion graph), shifted by p in the x-direction and by q in the y-direction.
An irrational function takes the form y=√(expression), with a variable inside the root sign. Since the value inside the root can never be negative, irrational functions have the distinctive feature of a domain restricted to only one direction. For example, y=√(x−p)+q requires x−p to be at least 0, so the domain is restricted to x≥p, and the graph starts at the point (p,q) and gradually curves up and to the right. Flip the sign inside the root or in front of it, and the direction the graph extends from its starting point changes to up, down, left, or right.
A composite function is a different kind of animal from the first two. Given two functions f and g, (f∘g)(x)=f(g(x)) means "feed x into g first, then feed that result into f." For example, if f(x)=2x+1 and g(x)=x−3, then (f∘g)(x)=f(g(x))=f(x−3)=2(x−3)+1=2x−5. The key thing to remember here is that swapping the order, (g∘f)(x)=g(f(x)), usually gives you a completely different function. Using the same example, (g∘f)(x)=g(2x+1)=(2x+1)−3=2x−2, which is a different expression from (f∘g)(x)=2x−5. A composite function isn't an operation you can freely reorder like multiplication.
An inverse function is a function with the input and output of some function f completely swapped. If f(x)=ax+b, swap the roles of x and y and solve for y again, and you get the inverse function f⁻¹(x)=(x−b)/a. For an inverse function to exist, the original function has to be one-to-one — "each output corresponds to exactly one input" (a linear function always satisfies this condition).
So reflecting any point (t, f(t)) on f across the line y=x always gives the point (f(t), t) on f⁻¹. Because of this property, the graphs of f and f⁻¹ are always mirror images of each other (more precisely, symmetric) about the line y=x. It's also worth remembering that the domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse function.
All four functions might look unfamiliar, but they actually share something in common. Rational and irrational functions build the habit of first asking, "what condition does x need to satisfy for this expression to be defined?" while composite and inverse functions build the mindset of "treating a function as a single unit, then combining it with others or flipping it." This way of thinking is a genuinely important tool you'll keep using in calculus later on.
On our activity page, you can use sliders to change the position of the asymptotes and the direction of the domain yourself, use a function-machine diagram to check the order of operations in a composite function, and watch on the graph how a point on an inverse function is reflected about y=x.