IBMaths: AA HLDPTopic Questions2. Functions2.9 Further Functions & Graphs

Further Functions & Graphs (DP IB Maths: AA HL)

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2.9 Further Functions & Graphs

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1a
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a)
Sketch the graph of y equals left parenthesis x minus 1 right parenthesis squared minus 2 vertical line x minus 1 vertical line minus 1 comma for negative 3 less or equal than x less or equal than 6.

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1b
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b)
Hence, solve the equationspace y equals left parenthesis x minus 1 right parenthesis squared minus 2 vertical line x minus 1 vertical line minus 1 equals 0.

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2
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Given that  

space f left parenthesis x right parenthesis equals ln space x comma space space space space space space space space space x greater than 0 

sketch on separate axes the graphs of 

(i)
y equals f left parenthesis x right parenthesis space

(ii)
y equals vertical line f left parenthesis x right parenthesis vertical line

(iii)
y equals negative f left parenthesis x minus 3 right parenthesis 

On each diagram, show the x-intercepts along with any asymptotes, including their equations.

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3a
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The graph of y equals f left parenthesis x right parenthesis is given below.

q3a_2-9_medium_ib-aa-hl-mathsOn separate axes, draw the graphs of 

a)
vertical line f left parenthesis x right parenthesis vertical line

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3b
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b)
stretchy left square bracket f stretchy left parenthesis x stretchy right parenthesis stretchy right square bracket squared

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4a
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a)
Sketch the curve begin mathsize 16px style y equals fraction numerator 3 over denominator x plus 4 end fraction end style and line y equals 4 minus x on the same axes, clearly indicating any x- and  y- intercepts and any asymptotes.

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4b
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b)
Consider the equation 

4 minus x equals open vertical bar fraction numerator 3 over denominator x plus 4 end fraction close vertical bar 

(i)
Explain why the cases x less than negative 4 comma x equals negative 4 and x greater than negative 4 must be considered separately in attempting to solve the equation. 

(ii)
Hence find the exact solutions to the equation.

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5a
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Consider the function space f defined byspace f left parenthesis x right parenthesis equals 3 x squared space arcsin space x, negative 1 less or equal than x less or equal than 1.

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

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5b
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b)
State the range ofspace f.

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5c
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c)
Solve the inequality  vertical line 3 x squared arcsin space x vertical line greater than 1.

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6a
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Consider the functionspace f defined byspace f left parenthesis x right parenthesis equals square root of 9 minus x end root, wherespace f has the largest possible valid domain.

a)
(i)
Sketch the graph of y equals f left parenthesis x right parenthesis, labelling the  x- and  y-intercepts. 

(ii)
State the domain and range ofspace f.

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6b
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b)
(i)
On the same set of axes, sketch the graph of the functionspace f left parenthesis vertical line x vertical line right parenthesis, labelling the x- and y-intercepts.

(ii)
State the domain and range of the functionspace f left parenthesis vertical line x vertical line right parenthesis.

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7a
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 Let begin mathsize 16px style space f left parenthesis x right parenthesis equals fraction numerator 7 minus 9 x over denominator c x minus 12 end fraction end style, begin mathsize 16px style x not equal to 12 over c end style,  where c is a non-zero constant. 

The line x equals 4 is a vertical asymptote to the graph of y equals f left parenthesis x right parenthesis. 

a)
(i)
Find the value of c.

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

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7b
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b)
The line y equals k comma where k element of straight real numbers comma intersects the graph of y equals vertical line f left parenthesis x right parenthesis vertical line at exactly one point. Find the possible values of k.

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8a
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Let f left parenthesis x right parenthesis equals 2 x cubed minus 2 x , for x element of straight real numbers  .

(a)
(i)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar.
 
(ii)
State the transformation of the graph y equals f left parenthesis x right parenthesis  to y equals open vertical bar f left parenthesis x right parenthesis close vertical bar for f left parenthesis x right parenthesis less than 0.

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8b
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(b)
(i)
Sketch the graph of y equals f left parenthesis open vertical bar x close vertical bar right parenthesis.

 

          (ii)    State the transformation of the graph y equals f left parenthesis x right parenthesis to  y equals f left parenthesis open vertical bar x close vertical bar right parenthesis  for x less than 0.

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9a
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Let f open parentheses x close parentheses equals x open parentheses x minus 2 close parentheses.

(a)
Sketch the graph of y equals f open parentheses x close parentheseson the coordinate axes below. Be sure to label anywhere the graph intersects the coordinate axes and any extrema.
q9_2-9_ib-aa-hl_further-functions--graphs_medium_dig

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9b
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(b)
On the same axes, sketch the graph of the reciprocal y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction.Be sure to label anywhere the graph intersects the coordinate axes and any extrema.

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9c
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(c)
Find the equation of the horizontal and vertical asymptotes of the graph of y equals f left parenthesis x right parenthesis.

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1a
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Consider the function f left parenthesis x right parenthesis equals x squared minus 4 vertical line x vertical line minus 5 comma space x element of R. 

(a)
Solve f left parenthesis x right parenthesis equals 0.

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1b
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(b)
Sketch the graph of f. Clearly indicate the intersections with the coordinate axes and any turning points.

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2
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Given that

f left parenthesis x right parenthesis equals e to the power of x comma x element of R comma 

sketch on separate axes the graphs of

(i)
y equals f left parenthesis vertical line x minus 1 vertical line right parenthesis
(ii)
y equals vertical line f left parenthesis x right parenthesis minus 1 vertical line
(iii)
y equals f left parenthesis negative vertical line x vertical line right parenthesis.

Show any intercepts with the axes, label any local maximum and minimum points and give the equations of any asymptotes. Leave numbers in terms of e where appropriate.

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3a
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Consider the function f defined by f left parenthesis x right parenthesis equals 4 x squared minus 8 x minus 5.

(a)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar. Clearly indicate any intercepts with the axes and any turning points.

 

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3b
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(b)
Sketch the graph of y equals open square brackets f open parentheses x close parentheses close square brackets squared. Clearly indicate any intercepts with the axes and any turning points.

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4a
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(a)
Sketch the curve y equals fraction numerator 4 over denominator x plus 3 end fraction and the line y equals 3 minus x on the same diagram, clearly indicating any x- and  y- axes intercepts as well as any asymptotes.

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4b
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(b)
Hence find the exact solutions to the equation
3 minus x equals open vertical bar fraction numerator 4 over denominator x plus 3 end fraction close vertical bar.

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5
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The graph of f has two asymptotes with equation x equals 0.25 and y equals 0.5 as shown below. The graph passes through the points straight A left parenthesis 0 comma 6 right parenthesis and straight B left parenthesis 3 comma space 0 right parenthesis.

q5-2-9-ib-aa-hl-further-functions-_-graphs-hard-dig

Sketch the graph of y equals fraction numerator 1 over denominator f open parentheses x close parentheses end fraction. Clearly indicate the points where the graph intersects the axes or has a discontinuity and state the equations of any asymptotes.

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6a
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Consider the function  defined by f open parentheses x close parentheses equals ln open parentheses k minus x close parentheses where f has the largest possible valid domain and k is a positive constant such that k greater than 1.

(a)
Sketch the graph of y equals f open parentheses x close parentheses. Give the equations of any asymptotes and any intercepts with the axes in terms of k. Clearly state the domain and range of f.

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6b
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The function g is defined by g open parentheses x close parentheses equals f open parentheses open vertical bar x close vertical bar close parentheses. The range of g is g open parentheses x close parentheses less or equal than 1.

(b)
(i)
Find the exact value of k.
(ii)
State the domain of g .
(iii)
Sketch the graph of g.

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6c
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(c)
Given that open vertical bar g open parentheses x close parentheses close vertical bar equals p  has exactly two distinct real solutions find the range of values of p.

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7
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Find the set of values of x which satisfy the inequality

open vertical bar 2 x squared minus 13 x plus 15 close vertical bar greater than 3 x minus 15

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8a
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(a)
By considering the inverse of an appropriate function, sketch the graph y equals square root of x .

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8b
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The function f is defined by f open parentheses x close parentheses equals square root of 16 minus 2 x end root and its domain is the largest possible set of real values. 

(b)      (i)     State the domain and range of f . 

(ii)    Sketch the graph of y equals f open parentheses x close parentheses. Clearly label the points where the graph intersects the axes.

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8c
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(c)
On separate sets of axes, sketch the graphs of:
(i)
y equals f open parentheses open vertical bar x close vertical bar close parentheses
(ii)
y equals open square brackets f open parentheses x close parentheses close square brackets squared.
 
For each graph, define the domain and range and clearly label the points where the graph intersects the axes.

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1a
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(a)
By considering graph transformations of an appropriate quadratic function, sketch the graph of y equals open parentheses x plus 4 close parentheses squared minus 2 open vertical bar x plus 4 close vertical bar minus 3 . Clearly indicate any x-intercepts and any
y-intercepts.

 

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1b
Sme Calculator4 marks
(b)
Hence, find the values of k such that the equation left parenthesis x plus 4 right parenthesis squared minus 2 vertical line x plus 4 vertical line minus 3 equals k has exactly 4 distinct real solutions.

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2a
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A function f is defined by f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x minus 3 right parenthesis comma space space x element of R. 

(a)
(i)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar.
(ii)
Hence find the set of values of x for which f open parentheses x close parentheses less than open vertical bar f open parentheses x close parentheses close vertical bar

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2b
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(b)
Solve the equation f left parenthesis x right parenthesis equals vertical line x squared minus 4 x plus 3 vertical line.

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3a
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A function f is defined by f left parenthesis x right parenthesis equals x squared minus 8 x plus 15 comma space x element of straight real numbers.

(a)
Sketch the graph of y equals fraction numerator 1 over denominator f open parentheses x close parentheses end fraction.  Clearly indicate any intercepts with the coordinate axes and state the equations of any asymptotes. Find the coordinates of any turning points.

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3b
Sme Calculator6 marks
(b)
By sketching the graphs of  y equals f open parentheses x close parentheses and y on the same axes, find the values of x  for which  f open parentheses x close parentheses less or equal than fraction numerator 1 over denominator f open parentheses x close parentheses end fraction.

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4a
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Consider the functionf open parentheses x close parentheses equals fraction numerator open parentheses x plus 3 close parentheses open parentheses x minus 4 close parentheses over denominator x minus 5 end fraction comma space x element of straight real numbers comma space x not equal to 5. spaceThe graph of f  is shown below.

q4a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

(a)
Find the equation of the oblique asymptote of the graph of f .

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4b
Sme Calculator5 marks
(b)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar. Clearly indicate the points where the graph crosses the coordinates axes and state the equations of the asymptotes.

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4c
Sme Calculator4 marks
(c)
Sketch the graph of y equals f open parentheses open vertical bar x close vertical bar close parentheses. Clearly indicate the points where the graph crosses the coordinates axes and state the equations of the asymptotes.

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5a
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The graph of a function f is shown below. The equations of the asymptotes are x equals 0, y=-1 and y equals 1.The point A(1, 2)  lies on the graph.

q5a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

On separate sets of axes, sketch the graphs defined below. For each sketch, clearly label any asymptotes or discontinuities and clearly show the coordinates of the point where A gets mapped to.

(a)
y equals fraction numerator 1 over denominator f open parentheses x close parentheses end fraction plus 2.

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5b
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(b)
y equals open square brackets f open parentheses x close parentheses minus 2 close square brackets squared.

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5c
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(c)
y equals 1 over open square brackets f open parentheses x close parentheses close square brackets squared

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6a
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Let f be the function defined by f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis x squared plus x minus 26 right parenthesis. 

(a)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar. Give the exact coordinates of the x-intercepts.

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6b
Sme Calculator6 marks
(b)
Use differentiation to find the set of values of k such that there are 4 points of intersection between the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar and the line y equals k.

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7a
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The graph of the function f left parenthesis x right parenthesis equals a vertical line x plus p vertical line plus q space spaceis shown below, where a comma space p space comma space q element of straight real numbers. The graph has a local maximum at the point A(3, 5)  and intersects the y-axis at (0,-7).

q7a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

 

(a)
Find the values of a comma space p and q.Hence solve f open parentheses x close parentheses equals 0.

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7b
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(b)
Find the solutions to the equation f left parenthesis x right parenthesis equals open vertical bar 7 minus 2 x close vertical bar.

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8a
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Consider the function defined by f left parenthesis x right parenthesis equals x vertical line x vertical line comma x element of R.

(a)
Sketch the graph of f.

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8b
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(b)
By first sketching the graph of f to the power of negative 1 end exponent  on the same set of axes as the graph of f, solve f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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8c
Sme Calculator3 marks
(c)
Given that k is a real constant, find an expression for f to the power of negative 1 end exponent open parentheses k close parentheses in the case when:
(i)
k greater or equal than 0 comma
(ii)
k less than 0.

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9a
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The graph of a function f is shown below.

q9a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

On separate sets of axes sketch the following graphs clearly showing any key points.

a)
y equals vertical line f left parenthesis x right parenthesis vertical line minus f left parenthesis x right parenthesis.

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9b
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(b)
y equals f left parenthesis x right parenthesis vertical line f left parenthesis x right parenthesis vertical line

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9c
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(c)
y equals fraction numerator f open parentheses x close parentheses over denominator open vertical bar f open parentheses x close parentheses close vertical bar end fraction

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10
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 f left parenthesis x right parenthesis equals x to the power of 4 minus 8 x cubed minus 40 x squared plus 224 x minus 240.

Find the values of  such that open vertical bar f open parentheses open vertical bar x close vertical bar close parentheses close vertical bar equals k has six distinct, real solutions.
Explain each stage of your solution in full.

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