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
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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 \(y=\frac{2x}{x-5}\)
x =
The vertical asymptote occurs when x - 5 = 0
What is the equation of the horizontal asymptote of the graph \(y=\frac{2x}{x-5}\)
y =
For the graph \(y=\frac{ax+b}{cx+d}\), the horizontal asymptote is \(y=\frac{a}{c}\)
What is the y intercept of the function \(f(x)=\frac{2x-4}{5x+1}\)
y =
The graph intersects the y axis when x = 0
\(\frac{0-4}{0+1}=-4\)
What is the x intercept of the function \(f(x)=\frac{2x-4}{5x+1}\)
x =
The graph intersects the x axis when y = 0
\(\frac{2x-4}{5x+1}=0\)
2x - 4 = 0
x = 2
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 \(y=-\frac{1}{0+1}=-1\)
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 \(y=\frac{0+ 2} {0 - 2}=-1\)
The function \(f(x) =\frac{ax+1}{2x+b}\) is plotted
What are the values of a and b
a =
b =
| 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 |
The function \(f(x) =\frac{ax+3}{bx+3}\) is plotted
What are the values of a and b
a =
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 |
The function \(f(x) =\frac{4x+1}{ax+b}\) is plotted
What are the values of a and b
a =
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 |
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
\(f(x)=\frac{4x+2}{2x+1},x\neq{-0.5}\)
\(f(x)=\frac{2(2x+1)}{2x+1},x\neq{-0.5}\)
\(f(x)=2,x\neq{-0.5}\)
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 \(y=\frac{x^2+1}{1-x}\)?
x =
vertical asymptote occurs where 1 - x = 0
What is the equation of the vertical asymptote in the graph of \(y=\frac{2x+1}{(x+2)^2}\)?
x =
vertical asymptote occurs where (x + 2)² = 0
What is the equation of the horizontal asymptote in the graph of \(y=\frac{1-x}{x^2+1}\)?
y =
Rational functions in the form \(\frac{ax+b}{cx^2+dx+e}\)have a horizontal asymptote at y = 0
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
y = , x =
The vertical asymptote occurs where \({2x+1}=0\)
The oblique asymptote occurs where y = x + 1
\(f(x) = x+a+\frac{2}{bx-1}\) has asymptotes at y = x - 1 and x = 0.5
Find the values of a and b.
a =
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
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.
a =
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
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
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 \(f(x)=\frac{4-x}{(x+2)(x-1)}\)
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
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
\(y=0-3+\frac{2}{0+2}=-2\)
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
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
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
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
How much of Rational Functions HL have you understood?
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