Reciprocal & Rational Functions | DP IB Maths: AA SL Revision Notes 2021

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Reciprocal Functions & Graphs

The reciprocal function is defined by $f (x) = \frac{1}{x}, x \neq 0$
Its domain is the set of all real values except 0
Its range is the set of all real values except 0
The reciprocal function has a self-inverse nature
- $f^{- 1} (x) = f (x)$
- $(f \circ f) (x) = x$

The graph does not have a y-intercept
The graph does not have any roots
The graph has two asymptotes
- A horizontal asymptote at the x-axis: $y = 0$
  - This is the limiting value when the absolute value of x gets very large
- A vertical asymptote at the y-axis: $x = 0$
  - This is the value that causes the denominator to be zero
The graph has two axes of symmetry
- $y = x$
- $y = - x$
The graph does not have any minimum or maximum points

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A rational function is of the form $f (x) = \frac{a x + b}{c x + d}, x \neq - \frac{d}{c}$
Its domain is the set of all real values except $- \frac{d}{c}$
Its range is the set of all real values except $\frac{a}{c}$
The reciprocal function is a special case of a rational function

The graph has a y-intercept at $(0, \frac{b}{d})$ provided $d \neq 0$
The graph has one root at $(- \frac{b}{a}, 0)$ provided $a \neq 0$
The graph has two asymptotes
- A horizontal asymptote: $y = \frac{a}{c}$
  - This is the limiting value when the absolute value of x gets very large
- A vertical asymptote: $x = - \frac{d}{c}$
  - This is the value that causes the denominator to be zero
The graph does not have any minimum or maximum points
If you are asked to sketch or draw a rational graph:
- Give the coordinates of any intercepts with the axes
- Give the equations of the asymptotes

If you draw a horizontal line anywhere it should only intersect this type of graph once at most
The only horizontal line that should not intersect the graph is the horizontal asymptote
- This can be used to check your sketch in an exam

The function $f$ is defined by $f (x) = \frac{10 - 5 x}{x + 2}$ for $x \neq - 2$ .

Write down the equation of

(i)

the vertical asymptote of the graph of

f

(ii)

the horizontal asymptote of the graph of

f

2-4-1-ib-aa-sl-rational-func-a-we-solution

Find the coordinates of the intercepts of the graph of

f

with the axes.

2-4-1-ib-aa-sl-rational-func-b-we-solution

Sketch the graph of

f

2-4-1-ib-aa-sl-rational-func-c-we-solution