Rational Functions HL

On this page, we will look at the properties of the reciprocal function and rational functions. You may be required to draw a sketch of these functions. In these cases, it is important to know how to find the vertical asymptote, the horizontal asymptote any oblique asymptote and the x and y intercepts.


Key Concepts

On this page, you should learn about 

  • the reciprocal function \(f(x)=\frac{1}{x}\)
  • rational functions of the form \(f(x)=\frac{ax+b}{cx+d}\)
  • equations of vertical and horizontal asymptotes
  • rational functions of the form \(f(x)=\frac{ax+b}{cx^2+dx+e}\)
  • rational functions of the form \(f(x)=\frac{ax^2+bx+c}{dx+e}\)
  • oblique asymptotes

Summary

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Test Yourself

Here is a quiz about rational functions in the form \(f(x)=\frac{ax+b}{cx+d}\)


START QUIZ!

Here is a second quiz about rational functions in the form \(f(x)=\frac{ax+b}{cx^2+dx+e}\) and \(f(x)=\frac{ax^2+bx+c}{dx+e}\)


START QUIZ!

Exam-style Questions

Question 1

Let f(x) = 2x + 1 and \(g(x)=\frac{x}{1-x} \ ,x\neq1\)

a) Show that \(f\circ g(x)=\frac{x+1}{1-x}\)

b) Let \(h(x)=\frac{x+1}{1-x}\) , for x < 1

c) Sketch the graph of h

d) Sketch the graph of \(h^{-1}\)

Hint

Full Solution

Question 2

Let \(f(x)=\frac{3x-2}{x-a},x\neq\ a\)

a) Find the inverse function \(f^{-1}(x)\) in terms of a

b) Find the value of a such that f is a self-inverse function

Hint

Full Solution

 

Question 3

The function f is defined by \(f(x)=\frac{6x+1}{2x-1},x\in\mathbb{R},x\neq\frac{1}{2}\)

a) Write f(x) in the form \(A+\frac{B}{2x-1}\) where A and B are constants

b) Sketch the graph of f(x) stating the equations of any asymptotes and the coordinates of any intercepts with the axes

Hint

Full Solution

 

Question 4

Sketch the graph of \(f\left(x\right)=\frac{x^2+x-1}{x+2}\) giving the equations of any asymptotes and the coordinates of the x and y intercepts as well as any stationary points


Hint

Full Solution

 

Question 5

Find the value(s) of a such that \(f(x)=\frac{x+1}{ax^2+3x+2}\) has only one vertical asymptote.

Hint

Full Solution

 

MY PROGRESS

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