IBMaths: AA HLDPTopic Questions2. Functions2.5 Reciprocal & Rational Functions

Reciprocal & Rational Functions (DP IB Maths: AA HL)

Topic Questions

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2.5 Reciprocal & Rational Functions

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1a
Sme Calculator3 marks

Let f left parenthesis x right parenthesis equals fraction numerator 2 x plus 1 over denominator x minus 4 end fraction comma x not equal to 4.

For the graph of f, find the equation of:

(i)

the vertical asymptote

(ii)
the horizontal asymptote.

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1b
Sme Calculator2 marks

Find f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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1c
Sme Calculator1 mark

Write down the equation of the vertical asymptote to the graph of f to the power of negative 1 end exponent open parentheses x close parentheses

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2
Sme Calculator6 marks

Consider f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator 3 x plus c end fraction , for x not equal to negative c over 3, where a comma b comma c element of straight integer numbers.

y equals negative 1  and  x equals 2  are the equations of the asymptotes of the graph of f. Point  A open parentheses negative 2 comma negative 2 over 3 close parentheses lies on the graph.

Find the values of a comma b and c.

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3a
Sme Calculator1 mark

Consider the function f defined by f left parenthesis x right parenthesis equals fraction numerator 3 over denominator x plus 5 end fraction minus 2 ,  x element of R ,  x not equal to p .

Write down the value of p.

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3b
Sme Calculator1 mark

Write down the equation of the horizontal asymptote to the graph of y equals f left parenthesis x right parenthesis.

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3c
Sme Calculator2 marks

Show that fraction numerator space 3 over denominator x plus 5 end fraction minus 2 equals fraction numerator a x plus b over denominator x plus 5 end fraction,  where a and b are constants to be determined.

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3d
Sme Calculator3 marks

Sketch the graph of y equals f left parenthesis x right parenthesis.

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4a
Sme Calculator3 marks

Let f left parenthesis x right parenthesis equals fraction numerator 4 x minus 2 over denominator 2 x plus 5 end fraction, for x not equal to negative 5 over 2.

For the graph of f, find the coordinates of

(i)

the x-intercept

(ii)
the y-intercept.

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4b
Sme Calculator3 marks

For the graph of f, find the equation of

(i)

the vertical asymptote

(ii)
the horizontal asymptote.

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5a
Sme Calculator2 marks

Consider the function f defined by  f left parenthesis x right parenthesis equals fraction numerator 2 open parentheses 3 x minus 1 close parentheses over denominator open parentheses x plus 3 close parentheses open parentheses x minus 2 close parentheses end fraction , x element of straight real numbers ,  x not equal to negative 3 comma 2 .

Find the coordinates of the points where the graph of y equals f left parenthesis x right parenthesis  intersects the coordinate axes.

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5b
Sme Calculator3 marks

Express f left parenthesis x right parenthesis as partial fractions.

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5c
Sme Calculator2 marks

Hence find the equation of the horizontal asymptote to the graph of y equals f left parenthesis x right parenthesis.

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6a
Sme Calculator2 marks

Consider the function  f left parenthesis x right parenthesis equals fraction numerator 4 x minus 12 over denominator x squared minus 4 x minus 5 end fraction  ,x element of straight real numbers  , x not equal to negative 1 comma 5

Find the coordinates of the points where the graph of  y equals f left parenthesis x right parenthesis intersects the

(i)
x-axis
 
(ii)
y-axis.

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6b
Sme Calculator3 marks

Write down the equations of

(i)
the vertical asymptotes

(ii)
the horizontal asymptote

to the graph of y equals f left parenthesis x right parenthesis.

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6c
Sme Calculator4 marks

By considering the value of f spacefor large positive and large negative values of x, sketch the graph of f.  Be sure to indicate clearly the points of intersection with the coordinate axes, as well as any asymptotes.

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7a
Sme Calculator3 marks

Consider the function f left parenthesis x right parenthesis equals fraction numerator x squared plus 5 x plus 6 over denominator x plus 1 end fraction, x element of straight real numbers , x not equal to negative 1 .

Find the coordinates of the points where the graph of y equals f left parenthesis x right parenthesis intersects the

(i)
x-axis,

(ii)
y-axis.

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7b
Sme Calculator1 mark

Write down the equation of the vertical asymptote to the graph of y equals f left parenthesis x right parenthesis.

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7c
Sme Calculator4 marks
(i)
Show that  fraction numerator x squared plus 5 x plus 6 over denominator x plus 1 end fraction equals x plus a plus fraction numerator b over denominator x plus 1 end fraction , where a and b are constants to be determined.
(ii)
Hence write down the equation of the oblique asymptote to the graph of    y equals f left parenthesis x right parenthesis       .

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7d
Sme Calculator3 marks

Sketch the graph of y equals f left parenthesis x right parenthesis. Be sure to indicate clearly the points of intersection with the coordinate axes, as well as any asymptotes.

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8a
Sme Calculator3 marks

Let f be a function defined by f left parenthesis x right parenthesis equals 3 x minus 4 plus fraction numerator 2 over denominator x minus 1 end fraction  , x element of straight real numbers , x not equal to p .

Write down

(i)
the value of p

(ii)
the equation of the vertical asymptote to the graph of y equals f left parenthesis x right parenthesis

(iii)
the equation of the oblique asymptote to the graph of y equals f left parenthesis x right parenthesis.

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8b
Sme Calculator2 marks

Show that f left parenthesis x right parenthesis can be written in the form fraction numerator a x squared plus b x plus c over denominator x minus 1 end fraction,  where a, b and c are constants to be determined.

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8c
Sme Calculator3 marks

Use an algebraic method to show that the graph of y equals f left parenthesis x right parenthesis spacedoes not cross the x-axis.

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8d
Sme Calculator3 marks

Sketch the graph of y equals f left parenthesis x right parenthesis and hence write down the range of the function f.

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1a
Sme Calculator1 mark

Let f open parentheses x close parentheses equals fraction numerator b over denominator x minus a end fraction plus 2, for x not equal to a.

The line x equals negative 3 is a vertical asymptote of the graph of f.

Write down the value of a.

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1b
Sme Calculator3 marks

The graph of f passes through the point straight A open parentheses 4 comma negative 2 close parentheses.

Find the value of b.

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1c
Sme Calculator2 marks

Find limit as x rightwards arrow infinity of f open parentheses x close parentheses.

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2a
Sme Calculator6 marks

Consider f open parentheses x close parentheses equals fraction numerator a x plus b over denominator c x plus 1 end fraction, where a comma space b comma space c element of straight integer numbers.

y equals 4 and  x equals negative 1 are the equations of the asymptotes of the graph of f. Point straight P open parentheses 5 comma begin inline style 9 over 2 end style close parentheses lies on the graph.

Find the values of a comma space b and c.

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2b
Sme Calculator4 marks

Sketch the graph of y equals f open parentheses x close parentheses.

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3a
Sme Calculator1 mark

Let f open parentheses x close parentheses equals fraction numerator 3 over denominator x minus a end fraction plus b, for x not equal to a. 

The line x equals 2 is a vertical asymptote of the graph of f. 

Write down the value of a.

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3b
Sme Calculator3 marks

The graph of f passes through the point straight A open parentheses 1 comma 5 close parentheses.

Find the value of b.

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3c
Sme Calculator2 marks

Find limit as x rightwards arrow infinity of f open parentheses x close parentheses.

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4a
Sme Calculator2 marks

Consider the function f defined by f open parentheses x close parentheses equals fraction numerator 3 x minus 7 over denominator x squared minus 6 x plus 5 end fraction, for x not equal to 1.5.

Find the coordinates where the graph of f intersects the coordinate axes.

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4b
Sme Calculator3 marks
(i)
Express f open parentheses x close parentheses as partial fractions.

(ii)
Hence, write down the equation of the horizontal asymptote.

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4c
Sme Calculator4 marks

Sketch the graph of y equals f open parentheses x close parentheses.

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5a
Sme Calculator6 marks

The graph of a function f is shown below. The equations of the asymptotes are x equals 0.25 and y equals 0.5. The graph crosses the coordinate axes at the points straight A left parenthesis 0 comma space m right parenthesis and straight B left parenthesis 5 comma 0 right parenthesis.

q5a-2-5-ib-aa-hl-reciprocal-_-rational-functions-hard-maths-diagram

Find an equation for f open parentheses x close parentheses in the form  fraction numerator a x plus b over denominator c x plus d end fraction, where a comma space b comma space c comma space d element of straight integer numbers.

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5b
Sme Calculator1 mark

Write down the value of m.

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6a
Sme Calculator3 marks

Consider the function f open parentheses x close parentheses equals fraction numerator 2 x squared minus x minus 15 over denominator x minus 2 end fraction comma space x element of straight real numbers comma space x not equal to 2.

Find the coordinates where the graph of f crosses the

(i)
x-axis,

(ii)
y-axis.

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6b
Sme Calculator4 marks

Find the equation of the oblique asymptote of the graph of f, giving the answer in the form y equals a x plus b where a comma b element of straight integer numbers..

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6c
Sme Calculator4 marks

Sketch the graph of y equals f open parentheses x close parentheses. Clearly indicate the asymptotes and give coordinates of the points where the graph intersects the axes.

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7
Sme Calculator6 marks

Consider the function f defined by f open parentheses x close parentheses equals fraction numerator a x plus b over denominator 2 x squared plus c x plus d end fraction comma where a comma space b comma space c comma space d element of straight integer numbers. The lines x equals negative 1 half and x equals 3 are vertical asymptotes of the graph of f as shown below. The graph crosses through the points straight A open parentheses 0 comma 4 close parentheses and straight B open parentheses 2 comma 0 close parentheses.

q7-2-5-ib-aa-hl-reciprocal-_-rational-functions-hard-maths-diagram

Find the values of a comma space b comma space c and d.

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8a
Sme Calculator2 marks

The graph of a function f is shown below. The equations of the asymptotes are x equals 3 and y equals 3 x minus 2. The graph crosses the y-axis at open parentheses 0 comma space minus 2.5 close parentheses.

q8a-2-5-ib-aa-hl-reciprocal-_-rational-functions-hard-maths-diagram

The function can be written as f open parentheses x close parentheses equals a x plus b plus fraction numerator 3 over denominator c x plus d end fraction, where a comma space b comma space c comma space d space element of straight integer numbers. 

Find the values of a comma space b comma space c and d.

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8b
Sme Calculator3 marks

Find the exact coordinates of the points where y equals f open parentheses x close parentheses crosses the x-axis.

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8c
Sme Calculator2 marks

Given that f open parentheses x close parentheses equals k has no real solutions where k element of straight real numbers, find the set of possible values of k. Give the bounds correct to 2 decimal places.

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9a
Sme Calculator4 marks

Consider the function f open parentheses x close parentheses equals fraction numerator 2 x squared minus 18 over denominator x squared minus 25 end fraction, for x not equal to plus-or-minus 5.

(i)
Show that fraction numerator 2 x squared minus 18 over denominator x squared minus 25 end fraction equals 2 plus fraction numerator 16 over denominator 5 open parentheses x minus 5 close parentheses end fraction minus fraction numerator 16 over denominator 5 open parentheses x plus 5 close parentheses end fraction.

(ii)
Hence, write down the value of limit as x rightwards arrow infinity of f open parentheses x close parentheses.

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9b
Sme Calculator4 marks
(i)
Sketch the graph of y equals f open parentheses x close parentheses.

(ii)
Write down the range of f.

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10a
Sme Calculator2 marks

Consider the function f open parentheses x close parentheses equals fraction numerator 2 x squared minus 72 over denominator x squared minus 9 end fraction, for x not equal to plus-or-minus 3. 

Prove that f is an even function.

 

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10b
Sme Calculator5 marks
(i)
Sketch the graph of y equals f open parentheses x close parentheses.

(ii)
Write down the range of f.

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1
Sme Calculator7 marks

The graph of a function f  is shown below. The graph crosses the x-axis at the point A and the y-axis at point B. The asymptotes intersect at the point Copen parentheses 4 comma space minus 8 over 3 close parentheses. The area of the triangle triangle AOB is 3.

q1-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

Find an equation for f open parentheses x close parentheses in the form fraction numerator a x plus b over denominator c x plus d end fraction, where a comma space b comma space c comma space d element of straight integer numbers. .

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2a
Sme Calculator1 mark

Let the function f be defined by f open parentheses x close parentheses equals 3 minus fraction numerator a over denominator 2 x minus 5 end fraction comma space x element of straight real numbers comma space x not equal to p, where a is a positive constant. 

Write down the value of p.

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2b
Sme Calculator4 marks

Sketch the graph of f . State the equations of the asymptotes and give the coordinates of the intersections with the coordinate axes in terms of a.

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2c
Sme Calculator4 marks

Given that the graph of f intersects the line given by y equals x, find the set of possible values of a.

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3a
Sme Calculator3 marks

Consider the function  defined by f open parentheses x close parentheses equals fraction numerator 4 minus 2 x over denominator x squared minus 3 x plus 3 end fraction.The graph of f is shown below. The graph has a maximum point at A and a minimum point at B.

q3a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

Show that f is defined for all x element of straight real numbers.

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3b
Sme Calculator6 marks
(i)
Given that the graph of f intersects the line y equals k, show that 3 k squared minus 4 k minus 4 less or equal than 0.

(ii)
Hence, find the range of f.

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3c
Sme Calculator4 marks

Hence, find the coordinates of A and B.

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4a
Sme Calculator5 marks

Consider the function f defined by f open parentheses x close parentheses equals fraction numerator open parentheses x minus p close parentheses open parentheses x minus q close parentheses over denominator x minus n end fraction, where n comma space p and qare positive constants and p less than q.

In the case that p comma space qand n are distinct:

(i)
write down the coordinates of the points where the graph of f intersects the axes,

(ii)
write down the equation of the vertical asymptote,

(iii)
show that the equation of the oblique asymptote is  y equals x plus left parenthesis n minus p minus q right parenthesis.

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4b
Sme Calculator6 marks

Sketch the graph of f in that case that:

(i)
n less than p less than q comma
(ii)
p less than n less than q,
(iii)
n equals p.

Clearly indicate where the graph crosses the coordinate axes and any asymptotes or discontinuities. 

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5a
Sme Calculator7 marks

Sketch the graph of the function f be defined by f open parentheses x close parentheses equals fraction numerator 4 x minus 1 over denominator 2 x squared minus x minus 6 end fraction. Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

 

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5b
Sme Calculator6 marks

Sketch the graph of the function begin mathsize 16px style g end style defined by gopen parentheses x close parentheses equals fraction numerator 2 x squared minus x minus 6 over denominator 4 x minus 1 end fraction  Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

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6a
Sme Calculator6 marks

The graph of a function f  is shown below. The graph crosses the x-axis at the points straight A open parentheses negative 0.5 comma space 0 close parentheses and straight B open parentheses 3 comma 0 close parentheses. The graph crosses the y-axis at straight C open parentheses 0 comma negative 3 close parentheses. The equation of the vertical asymptote is x equals 2.

q6a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

The function f can be written in the form f open parentheses x close parentheses equals fraction numerator a x squared plus b x plus c over denominator d x plus e end fraction, where a comma space b comma space c comma space d comma space e element of straight integer numbers. 

Given that c e plus 2 equals b , find the values of a comma space b comma space c comma space d spaceand e.

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6b
Sme Calculator4 marks

Hence find the equation of the oblique asymptote.

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7a
Sme Calculator8 marks

Consider the function f defined by f open parentheses x close parentheses equals fraction numerator a x plus b over denominator x squared plus c x plus d end fraction,  where a comma space b comma space c comma space d element of straight integer numbers. The line x equals 2 is the only vertical asymptote of the graph of f as shown below. The graph crosses through the points straight A open parentheses 0 comma k close parentheses and straight B open parentheses k comma space 0 close parentheses where k is a positive constant. The line y equals negative 1 is a tangent to the graph of f at the point straight C.

q7a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

Find the values of a comma space b comma space c comma space d space and k.

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7b
Sme Calculator2 marks

Find the coordinates of straight C.

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8a
Sme Calculator2 marks

Consider the function f open parentheses x close parentheses equals fraction numerator 2 x squared plus 5 x minus 12 over denominator x squared minus 4 end fraction comma space x element of straight real numbers comma space x not equal to plus-or-minus 2.

Solve f open parentheses x close parentheses equals 0

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8b
Sme Calculator3 marks

Show that fraction numerator 2 x squared plus 5 x minus 12 over denominator x squared minus 4 end fraction equals A plus fraction numerator B x plus C over denominator x squared minus 4 end fraction, where A comma space B and C are constants to be found.

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8c
Sme Calculator6 marks

Using the answer to part (b), sketch the graph of f. Clearly indicate the coordinates of the points where the graph intersects the axes and state the equations of any asymptotes.

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9a
Sme Calculator3 marks

The diagram below shows the graphs of two linear functions f and g and a quadratic function h.

q9a-2-5-ib-aa-hl-reciprocal-_-rational-functions-very-hard-maths-diagram

Sketch the graph of y equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction . Clearly indicate where the graph intersects the x-axis and the location of any vertical asymptotes.

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9b
Sme Calculator4 marks

Sketch the graph of y equals fraction numerator g open parentheses x close parentheses over denominator h open parentheses x close parentheses end fraction. Clearly indicate where the graph intersects the x-axis and the location of any vertical asymptotes.

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