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.
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 Here is a quiz about rational functions in the form \(f(x)=\frac{ax+b}{cx+d}\)
START QUIZ! What is the equation of the vertical asymptote of the graph \(y=\frac{2x}{x-5}\)
The vertical asymptote occurs when x - 5 = 0
Check
What is the equation of the horizontal asymptote of the graph \(y=\frac{2x}{x-5}\)
For the graph \(y=\frac{ax+b}{cx+d}\) , the horizontal asymptote is \(y=\frac{a}{c}\)
Check
What is the y intercept of the function \(f(x)=\frac{2x-4}{5x+1}\)
The graph intersects the y axis when x = 0
\(\frac{0-4}{0+1}=-4\)
Check
What is the x intercept of the function \(f(x)=\frac{2x-4}{5x+1}\)
The graph intersects the x axis when y = 0
\(\frac{2x-4}{5x+1}=0\)
2x - 4 = 0
x = 2
Check
Which of the following graphs represents the function \(f(x)=-\frac{1}{x+1}\)
\(f(x)=-\frac{1}{x+1}\) has a vertical asymptote at x = -1
The y intercept is at \(y=-\frac{1}{0+1}=-1\)
Check
Which of the following graphs represents the function \(f(x)=\frac{3x + 2} {2x - 2}\)
\(f(x)=\frac{3x + 2} {2x - 2}\) has
a horizontal asymptote at y = \(\frac{3}{2}\)
a vertical asymptote where 2x - 2 = 0, that is x = 1
a y intercept at \(y=\frac{0+ 2} {0 - 2}=-1\)
Check
The function \(f(x) =\frac{ax+1}{2x+b}\) is plotted
What are the values of a and b
The function
\(f(x) =\frac{ax+1}{2x+b}\) has
a vertical asymptote at \(x=-\frac{b}{2}\) a horizontal asymptote at \(y=\frac{a}{2}\) \(-\frac{b}{2}=2\) \(\frac{a}{2}=\frac{1}{2}\) b = - 4 a = 1
Check
The function \(f(x) =\frac{ax+3}{bx+3}\) is plotted
What are the values of a and b
The function \(f(x) =\frac{ax+3}{bx+3}\) has
a vertical asymptote at \(x=-\frac{3}{b}\) a horizontal asymptote at \(y=\frac{a}{b}\) \(-\frac{3}{b}=-3\) \(\frac{a}{b}=-2\) b = 1 a = - 2
Check
The function \(f(x) =\frac{4x+1}{ax+b}\) is plotted
What are the values of a and b
The function \(f(x) =\frac{4x+1}{ax+b}\) has
a horizontal asymptote at \(y=\frac{4}{a}\) a vertical asymptote at \(x=-\frac{b}{a}\) \(\frac{4}{a}=2\) \(-\frac{b}{2}=-\frac{5}{2}\) a = 2 b = 5
Check
Which of the following correctly describes this function
This is a special rational function that has a hole at x = - 0.5
Hence the denominator of the rational function is 2x + 1
\(f(x)=\frac{4x+2}{2x+1},x\neq{-0.5}\)
\(f(x)=\frac{2(2x+1)}{2x+1},x\neq{-0.5}\)
\(f(x)=2,x\neq{-0.5}\)
Check
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! What is the equation of the vertical asymptote in the graph of \(y=\frac{x^2+1}{1-x}\) ?
vertical asymptote occurs where 1 - x = 0
Check
What is the equation of the vertical asymptote in the graph of \(y=\frac{2x+1}{(x+2)^2}\) ?
vertical asymptote occurs where (x + 2)² = 0
Check
What is the equation of the horizontal asymptote in the graph of \(y=\frac{1-x}{x^2+1}\) ?
Rational functions in the form \(\frac{ax+b}{cx^2+dx+e}\) have a horizontal asymptote at y = 0
Check
What are the equations of the asymptotes of the function \(f(x)=x+1+\frac{3}{2x+1}\) ?
x + 1 -0.5 1.5 2x + 1
The vertical asymptote occurs where \({2x+1}=0\)
The oblique asymptote occurs where y = x + 1
Check
\(f(x) = x+a+\frac{2}{bx-1}\) has asymptotes at y = x - 1 and x = 0.5
Find the values of a and b .
\(f(x) = x+a+\frac{2}{bx-1}\) has asymptotes at y = x + a and where bx - 1 = 0
Hence, a = 1, b = 2
Check
The rational function \(f(x)=\frac{2x^2-x+b}{x-1}\) can be written as \(f(x)=ax+1+\frac{3}{x-1}\)
Work out a and b .
\(ax+1+\frac{3}{x-1}\equiv \frac{(ax+1)(x-1)+3}{x-1}\)
\(\equiv\frac{ax^2-ax+x-1+3}{x-1}\)
\(\equiv\frac{ax^2+(1-a)x+2}{x-1}\)
\(\equiv\frac{2x^2-x+b}{x-1}\)
Therefore, a = 2 and b = 2
Check
Which of the following is the graph of \(f(x)=\frac{x-2}{(x+1)(x-4)}\)
\(f(x)=\frac{x-2}{(x+1)(x-4)}\) has asymptotes at y = 0 , x = -1 and x = 4. This eliminates D
The x intercept is when x - 2 = 0, that is x = 2. This eliminates C
The y intercept is when \(\frac{0-2}{(0+1)(0-4)}=\frac{1}{2}\)
Hence the answer is B
Check
Which of the following is the correct rational function for the graph below
The graph has asymptotes at y = 0 , x = -2 and x = 1
The x intercept is at x = 4
Hence the correct function is \(f(x)=\frac{4-x}{(x+2)(x-1)}\)
Check
Which of the following is the graph of \(f(x)=x-1+\frac{2}{x+1}\)
\(f(x)=x-1+\frac{2}{x+1}\) has asymptotes at y = x - 1 and x = -1. This eliminates B
The y intercept is \(y=0-1+\frac{2}{0+1}=1\) . This eliminates C and D.
The correct answer is A
Check
Which of the following is the correct rational function for the graph below
The graph has asymptotes at y = x - 3 and x = -2
The y intercept is at y = -2
The correct answer is \(f(x)=x-3+\frac{2}{x+2}\) , since
\(y=0-3+\frac{2}{0+2}=-2\)
Check
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 c) & d) In a sketch of a graph include
the equations of the asymptotes x intercept y intercept Full Solution
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 If f is a self inverse function, then \(f(x)=f^{-1}(x)\) Full Solution
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 The horizontal asymptote is at x = A Full Solution
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 Before trying to sketch this function
Find the equation of the vertical asymptote Check for 'the hole' by looking for common factpr in numerator and denominator Find the equation of the oblique asymptote Find the y intercept Find any x intercepts Find any stationary points Full Solution
Find the value(s) of a such that \(f(x)=\frac{x+1}{ax^2+3x+2}\) has only one vertical asymptote.
Hint If f(x) has only one vertical asymptote, there are two possibilities
1) ax² + 3x + 2 has 2 equal factors
2) the numerator and denominator have a common factor
Full Solution MY PROGRESS
Self-assessment How much of Rational Functions HL have you understood?
My notes
Which of the following best describes your feedback?