On this page, we will look at the properties of the reciprocal function and the rational function. 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 and the x and y intercepts.
On this page, you should learn about
- the reciprocal function \(f(x)=\frac{1}{x}\)
- the rational function \(f(x)=\frac{ax+b}{cx+d}\)
- equations of vertical and horizontal asymptotes
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Here is a quiz that practises the skills from this page
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
\(f(x)=\frac{4x+2}{2x+1},x\neq{-0.5}\)
\(f(x)=\frac{2(2x+1)}{2x+1},x\neq{-0.5}\)
\(f(x)=2,x\neq{-0.5}\)
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
How much of Rational Functions SL have you understood?
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