IBMaths: AA SLDPTopic Questions2. Functions2.4 Further Functions & Graphs

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

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2.4 Further Functions & Graphs

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1a
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Consider the functions f open parentheses x close parentheses space equals space minus x to the power of 5 space plus 2020 and g open parentheses x close parentheses space equals space 1 over square root of open parentheses 1 minus x close parentheses end root cubed minus 2.

Find the coordinates of the y-intercepts for the graph of

(i)
f

(ii)
g.

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1b
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Find the coordinates of the x-intercepts for the graph of

(i)
f

(ii)
g.

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1c
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For the graph of g, find the equation of

(i)
the vertical asymptote

(ii)
the horizontal asymptote.

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2a
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Let f open parentheses x close parentheses equals fraction numerator 2 x space plus space 1 over denominator x space minus space 4 end fraction comma space x not equal to 4. space

For the graph of f, find the equation of:

(i)
the vertical asymptote

(ii)
the horizontal asymptote.

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2b
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Findspace f to the power of negative 1 end exponent left parenthesis x right parenthesis space.

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2c
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Write down the equation of the vertical asymptote to the graph of f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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3a
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Let f open parentheses x close parentheses equals ln space open parentheses x plus 2 close parentheses space comma space space x greater than negative 2.

Find the coordinates of:

(i)
the x-intercept

(ii)
the y-intercept.

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3b
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State the equation of the vertical asymptote to the graph of f.

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3c
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The graph of space y equals f open parentheses x close parentheses spaceintersects with its inverse, twice. 

Find the two coordinates wherespace f open parentheses x close parentheses equals f to the power of negative 1 end exponent left parenthesis x right parenthesis.

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4a
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Letspace f open parentheses x close parentheses equals 0.5 e squared to the power of x plus 1,  for negative 1 less or equal than x less or equal than 2.

On the following grid, sketch the graph of y equals f open parentheses x close parentheses.

q4a-2-4--furtherfunctions-graphs-medium-ib-aa-sl-maths

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4b
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The inverse of f can be written in the form of space f to the power of negative 1 end exponent open parentheses x close parentheses equals A ln space b open parentheses x minus c close parentheses space.
Find the values of A, b and of c.

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5a
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Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years.

 A model for the mass of carbon-14, m g, in an object of age t years is

m equals m subscript 0 e to the power of negative k t end exponent

where m subscript 0 and k are constants.

For an object initially containing 100g of carbon-14, write down the value of m subscript 0.

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5b
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Briefly explain why, if m subscript 0 equals 100,m  will equal 50g  when t equals 5700 years.

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5c
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Using the values from part (b), show that the value of k spaceis 1.22 cross times 10 to the power of negative 4 end exponent to three significant figures.

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5d
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A different object currently contains 60g of carbon-14.
In 2000 years’ time how much carbon-14 will remain in the object?

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6a
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A small company makes a profit of £2500 in its first year of business and £3700 in the second year.  The company decides they will use the model

P equals P subscript 0 space y to the power of k

to predict future years’ profits.

£P is the profit in the y to the power of t h end exponent year of business.

P subscript 0 and k are constants.

Write down two equations connecting P subscript 0 and k.

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6b
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Find the values of P subscript 0 and k.

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6c
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Find the predicted profit for years 3 and 4.

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6d
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Show that

P equals P subscript 0 space y to the power of k

can be written in the form

log space P equals log space P subscript 0 plus k space log space y

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7a
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In an effort to prevent extinction scientists released some rare birds into a newly constructed nature reserve.

The population of birds, within the reserve, is modelled by

B equals 16 e to the power of 0.85 t end exponent

B is the number of birds after t years of being released into the reserve.

Write down the number of birds the scientists released into the nature reserve.

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7b
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According to this model, how many birds will be in the reserve after 3 years?

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7c
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How long will it take for the population of birds within the reserve to reach 500?

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8a
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Rebecca recently had the COVID-19 vaccine. The volume, V, of the vaccine in her blood over time can be modelled by an equation of the form V subscript 1 open parentheses t close parentheses equals 1.7 t e to the power of negative 1.25 t end exponent, where V is the concentration (in mg) of the vaccine in the bloodstream and t is time measured in days after 9am on Monday.

On the following grid, sketch the graph of y equals V subscript 1 open parentheses t close parentheses.

q4a-2-4--furtherfunctions-graphs-medium-ib-aa-sl-maths

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8b
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Find, to the nearest minute, the time when the vaccine volume, V1 reaches a maximum value.

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8c
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Rebecca experienced side-effects from the vaccine between the times when the volume reached its maximum value until it had dropped to half of its maximum value. Find, to the nearest minute, the length of time that Rebecca experienced side-effects from taking the vaccine.

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8d
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The vaccine is medically determined to be no longer in Rebecca’s bloodstream when it drops down to 1% of its maximum value. Find the time that the vaccine is no longer in Rebecca’s bloodstream.

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8e
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Rebecca’s friend, Zara, also had the vaccine on the same day. The volume in Zara’s bloodstream can be modelled by an equation of the form of V subscript 2 open parentheses t close parentheses equals 1.766 t e to the power of negative 1.3 t end exponent. Calculate, to the nearest minute, how much faster V2. took to reach a maximum volume compared to V1.

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9a
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Let f open parentheses x close parentheses equals fraction numerator negative 3 over denominator x space minus space 3 end fraction, for x not equal to 3. 

For the graph of f, find:

(i)
the x- intercept

(ii)
the y- intercept

(iii)
the range of  f.

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9b
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Find the value of f to the power of negative 1 end exponent left parenthesis negative 1 right parenthesis.

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9c
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Given that g open parentheses x close parentheses equals f open parentheses x plus 3 close parentheses plus 1, find the domain and range of g.

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1a
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Let f left parenthesis x right parenthesis equals fraction numerator 7 over denominator 2 left parenthesis x minus 7 right parenthesis end fraction space minus 5, for x not equal to 7.

For the graph of f, find the:

(i)
x-intercept

(ii)
y-intercept.

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1b
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For the graph of f, write down the equation of any asymptotes.

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1c
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Let g left parenthesis x right parenthesis equals 2 left parenthesis 1 minus 2 x right parenthesis comma spacefor x element of straight real numbers. The graphs of f and g intersect at points straight P and straight Q.

Write down the coordinates of P and Q.

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1d
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Find the distance of PQ.

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2a
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Consider the function f left parenthesis x right parenthesis equals a left parenthesis 0.75 right parenthesis to the power of x plus b where a and b are constants. The graph of f passes through the points left parenthesis 0 comma 18 right parenthesis space and   left parenthesis 2 comma 11 right parenthesis  and is shown below.

l7Mb5_Ct_q4a-2-2-hard-ib-ai-sl-maths

Write down two equations relating a spaceand b.

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2b
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Find the value of a spaceand b.

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2c
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Write down the equation of the horizontal asymptote of the graph of f.

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3a
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A function is defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator left parenthesis x minus 3 right parenthesis squared end fraction plus 2 comma space x not equal to p.

Find the value of p.

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3b
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For the graph of f write down the equation of any asymptotes.

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3c
Sme Calculator1 mark

Find the range of f.

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3d
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The line  intersects the graph of f when x equals 1 and when x equals 4.

Find the equation of l. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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4a
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A function is defined by f left parenthesis x right parenthesis equals 4 minus space fraction numerator 12 over denominator left parenthesis 5 x plus 9 right parenthesis end fraction comma space x not equal to a.

Find the value of a. Give your answer as a fraction.

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4b
Sme Calculator3 marks

Find the range of f.

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4c
Sme Calculator2 marks

Find the value of f to the power of negative 1 end exponent left parenthesis 2 right parenthesis. Give your answer as a fraction.

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5a
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Let space f open parentheses x close parentheses equals fraction numerator 4 over denominator 6 minus x end fraction space, for x not equal to 6.

For the graph of f, find 

(i)
the x-intercept

(ii)
the y-intercept

(iii)
The equation of the vertical asymptote.

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5b
Sme Calculator4 marks

Let g open parentheses x close parentheses equals negative x over 4, for x element of straight real numbers. The graphs of f and g intersect at points A and B.

Find the coordinates of A and B.

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6a
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The average fat-free mass, M, in kg, of footballers as a function of their age, a, in years, can be given by the logarithmic function:

straight M open parentheses a close parentheses equals 10 log open parentheses a minus 15 close parentheses plus 50 comma space space space space space space space space space space space space space space space space 16 less or equal than a less or equal than 25.

Calculate the average fat free mass of players aged:

(i)
16 years

(ii)
25 years.

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6b
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Find an expression for a linear model using your answers to part (a) (i) and (ii).

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7a
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The number of bacteria, n, in a dish, after t minutes is given by n space equals 5231 e to the power of 0.12 t end exponent.

Find the initial amount of bacteria.

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7b
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Find the amount of bacteria after 12 minutes. Give your answer in the form a cross times 10 to the power of k, where 1 less or equal than a less than 10 comma space k element of straight integer numbers

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7c
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Find the value of t when n equals 2.7 cross times 10 to the power of 4.

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8a
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Let space f open parentheses x close parentheses equals e to the power of negative x end exponent plus 1 spaceand g open parentheses x close parentheses equals 2 x minus m, forspace x element of straight real numbers, where m is a constant.

Findspace left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis.

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8b
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Given that limit as x rightwards arrow infinity of open parentheses g ring operator f close parentheses open parentheses x close parentheses equals negative 1 , find the value of m.

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1a
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A function is defined by f left parenthesis x right parenthesis equals e to the power of x squared plus b x plus 4 end exponent. The graph of f  has an axis of symmetry of x equals 2.

Find the value of b.

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1b
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Find the range of f .

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1c
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Another function is defined by g open parentheses x close parentheses space equals space minus fraction numerator open parentheses x squared minus 25 close parentheses over denominator 5 end fraction. The graph of f and g intersect at points A and B.

Find the equation of the line passing through points A and B. Give your answer in the form y equals m x plus c.

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1d
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Find the distance of the line AB.

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2a
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Consider the function f left parenthesis x right parenthesis equals 5 minus log left parenthesis 6 minus 4 x right parenthesis. The line l subscript 1 intersects the graph of f at point Aleft parenthesis negative 1 comma y right parenthesis and Bleft parenthesis x comma 5 right parenthesis.

Find the value of x and y.

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2b
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Find the equation of l subscript 1. Give your answer in the form y equals m x plus c, where m and c are fractions.

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3a
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The function f is a quadratic in the form f left parenthesis x right parenthesis equals a x squared plus b x minus 2, for negative 10 less or equal than x less or equal than 10.

The graph of f has x-intercepts open parentheses fraction numerator 1 plus square root of 5 over denominator 2 end fraction comma space 0 close parentheses and open parentheses fraction numerator 1 minus square root of 5 over denominator 2 end fraction comma space 0 close parentheses.

Find the values of a and b.

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3b
Sme Calculator2 marks

Another function can be defined by g left parenthesis x right parenthesis equals 6 open parentheses 0.8 close parentheses to the power of negative x end exponent minus 1, for negative 10 less or equal than x less or equal than 10.

The graph of f and g intersect at points straight A and straight B.

Find the coordinates of straight A and straight B.

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3c
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Solve the inequality f left parenthesis x right parenthesis less than g left parenthesis x right parenthesis.

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4a
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Write down the domain and range of the logarithmic functionspace y equals log subscript b x space , where space b greater than 0 spaceand b not equal to 1.

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4b
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Given that log subscript y squared end subscript space x space equals 16 log subscript x open parentheses y squared space close parentheses, find all the expressions for x in terms of y.

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5a
Sme Calculator3 marks

Let f left parenthesis x right parenthesis equals 2 x to the power of 4 minus 2 x cubed minus 4 x squared plus x plus 1 where x space element of space straight real numbers.

Solve the inequality f left parenthesis x right parenthesis less than 0.

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5b
Sme Calculator3 marks

For the graph of f, find the coordinates of the

(i)
local maximum point.

(ii)
local minimum points.

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5c
Sme Calculator3 marks

Write down the possible domains of f for which f has an inverse and explain why the domain must be restricted.

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6a
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The following diagram shows part of the graph of  f  which crosses the x-axis at point A, with coordinates ( a, 0).  The line L is the tangent to the graph of   f   at the point B. 

q7a-2-4-further-functions-graphs-very-hard-ib-aa-sl-maths

Find the exact value of a.

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6b
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The x-coordinate of B is 10. The y-coordinate of B can be written in the form p ln space q , where p comma space q element of straight integer numbers.

Find the value of p and the value of q.

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6c
Sme Calculator5 marks

The gradient of L is 5 over 9 . The equation of L can be written in the form 

y equals 5 over 9 x minus u left parenthesis v minus ln space w right parenthesis space.

Find the values of u, v and w.

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7a
Sme Calculator3 marks

A population of endangered birds, P, can be modelled by the equation

P subscript t equals P subscript 0 e to the power of k t end exponent comma

where P0 is the initial population and t is measured in years. 

After three years, it is estimated that P subscript 3 over P subscript 0 space equals 0.87.

Find the value of k and interpret its meaning.

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7b
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Find the least number of whole years for which P subscript t over P subscript 0 space less than 0.45.

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8a
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The intensity of light, I, is assumed to be 100% at the surface of the ocean and decreases with depth, d, and can be estimated by the function

I open parentheses d close parentheses equals k open parentheses 1.08 close parentheses to the power of negative d end exponent

where I is expressed as a percentage, d is the depth below the surface, in metres, and k is a constant. 

Calculate the value of k.

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8b
Sme Calculator2 marks

State the domain and range of I.

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8c
Sme Calculator2 marks

Calculate the intensity of light 6.2 m below the surface.

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9a
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Let f open parentheses x close parentheses equals fraction numerator 6 minus 8 x over denominator c x minus 12 end fraction , for x not equal to 12 over c , where c not equal to 0.

The line x=2 is a vertical asymptote to the graph of y equals f left parenthesis x right parenthesis.

(i)
Find the value of c.

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

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9b
Sme Calculator3 marks

The linespace y equals h, where h element of straight real numbers, intersects the graph of y equals vertical line f open parentheses x close parentheses vertical line at exactly one point. Find possible values of h.

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