GCSEFurther MathsAQAExam Questions2. Algebra & FunctionsFunctions

Functions (AQA GCSE Further Maths): Exam Questions

Exam code: 8365

1 hour16 questions
Medium
Hard
Very Hard
1
Sme Calculator2 marks

The function   straight f open parentheses x close parentheses equals x squared minus 4 x plus 1   has domain  negative 1 less-than or slanted equal to x less-than or slanted equal to 5   

Here is the graph of   y equals straight f open parentheses x close parentheses

Graph of a parabola opening upwards on a grid, vertex at point (2, -3), with scales marked on both axes from -3 to 6.

Write down the range of straight f open parentheses x close parentheses for domain negative 1 less-than or slanted equal to x less-than or slanted equal to 5  

How did you do?

2a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals 3 x squared plus 6           for all x

straight g left parenthesis x right parenthesis equals square root of x minus 5 end root space space space space space space space space space space space x space greater-than or slanted equal to space 5

Work out the value ofspace gf left parenthesis 4 right parenthesis.

How did you do?

2b
Sme Calculator2 marks

Show that fg left parenthesis x right parenthesis can be written in the form a left parenthesis x – a right parenthesis where a is an integer.

How did you do?

3
Sme Calculator2 marks

straight f open parentheses x close parentheses equals x cubed minus 2

The domain of straight f left parenthesis x right parenthesis is   x less or equal than 3

Work out the range of straight f left parenthesis x right parenthesis.

How did you do?

4
Sme Calculator4 marks

straight f left parenthesis x right parenthesis equals 14 minus x squared for all real values of x.

Solve straight f left parenthesis 2 x right parenthesis equals 5

You must show your working.

How did you do?

5
Sme Calculator4 marks

On the grid, draw the graph of  y equals straight f left parenthesis x right parenthesis

table row cell straight f left parenthesis x right parenthesis end cell equals cell x plus 4 space space space space space space space space space space – 4 less-than or slanted equal to x less than 0 space end cell row blank equals cell 4 – 3 x space space space space space space space space space space space space 0 less-than or slanted equal to x less than 2 space end cell row blank equals cell – 2 space space space space space space space space space space space space space space space space space 2 less-than or slanted equal to x less-than or slanted equal to 5 end cell end table

Cartesian graph with x-axis from -4 to 5 and y-axis from -2 to 4, labelled at each integer. Gridlines form squares; origin marked as 'O'.

How did you do?

1
Sme Calculator4 marks

A function f is given by

table row cell straight f left parenthesis x right parenthesis end cell equals cell 4 x space space space space space space space space space space space space space space space space space space space x less than 0 space end cell row blank equals cell x squared – 8 x space space space space space space space space space space space space 0 less-than or slanted equal to x less-than or slanted equal to 8 space end cell row blank equals cell 16 – 2 x space space space space space space space space space space space space x greater than 8 end cell end table

A sketch of space y equals straight f left parenthesis x right parenthesis spaceis shown.

The line y=4x ends at the origin. It is joined by a parabola with vertex at (4,-16) that ends at (8,0). This is followed by a line with negative gradient.

Work out all the values of x for which  space straight f left parenthesis x right parenthesis equals negative 12.

How did you do?

2
Sme Calculator4 marks

A function is given by 

  table row cell space straight f left parenthesis x stretchy right parenthesis space end cell equals cell negative 2 x space space space space space space space space space space space space minus 1 less-than or slanted equal to x less than 0 space space end cell row cell space space space space space space space space space end cell equals cell x left parenthesis 4 – x right parenthesis space space space space space space space space space space space space 0 less-than or slanted equal to x less than 3 space end cell row cell space space space space space space space space space end cell equals cell 2 x – 3 space space space space space space space space space space space space space space space 3 less-than or slanted equal to x less-than or slanted equal to 4 end cell end table

Draw the graph of space y equals straight f left parenthesis x right parenthesis on the grid.

Graph with x-axis from -1 to 4 and y-axis from -1 to 5, displaying a grid for plotting points on the Cartesian plane.

How did you do?

3
Sme Calculator6 marks

straight f open parentheses x close parentheses equals fraction numerator x minus 3 over denominator 2 x end fraction

Solve space straight f left parenthesis x plus 1 right parenthesis – straight f left parenthesis 2 x right parenthesis equals 0.5

You must show your working.

How did you do?

4
Sme Calculator2 marks

straight g open parentheses x close parentheses equals 5 minus x squared

The domain of straight g open parentheses x close parentheses is – 2 less-than or slanted equal to x less-than or slanted equal to 1

Work out the range of straight g open parentheses x close parentheses.

How did you do?

5a
Sme Calculator1 mark

straight f left parenthesis x right parenthesis equals x squared minus 7     for all values of x

straight g left parenthesis x right parenthesis equals 1 – 3 x     for – 4 less-than or slanted equal to x less-than or slanted equal to 4

Work out the range of straight f left parenthesis x right parenthesis.

Give your answer as an inequality.

How did you do?

5b
Sme Calculator2 marks

Work out the range of straight g left parenthesis x right parenthesis.

Give your answer as an inequality.

How did you do?

5c
Sme Calculator4 marks

Solve 2 straight f left parenthesis x right parenthesis =gleft parenthesis x right parenthesis

You must show your working.

Give your answers to 3 decimal places.

How did you do?

6
Sme Calculator5 marks

straight f open parentheses x close parentheses equals fraction numerator x over denominator 2 x plus 1 end fraction  for positive values of x.

Work out   straight f open parentheses x plus 1 close parentheses minus straight f open parentheses x close parentheses

Give your answer as a fraction in its simplest form. You must show your working.

How did you do?

7a
Sme Calculator1 mark

The function straight f is given by    straight f left parenthesis x right parenthesis equals 3 x – 5

The range is 13 less than straight f left parenthesis x right parenthesis less than 19

Work out the domain of the function.

How did you do?

7b
Sme Calculator2 marks

The function straight g is given by straight g left parenthesis x right parenthesis equals x squared – 4 with domain – 1 less than x less than 3

Work out the range of the function.

How did you do?

7c
Sme Calculator2 marks

The function h is given by straight h left parenthesis x right parenthesis equals fraction numerator 3 plus x over denominator 2 end fraction

Work outspace straight h to the power of negative 1 end exponent left parenthesis x right parenthesis.

How did you do?

1
Sme Calculator5 marks

straight f left parenthesis x right parenthesis space equals space open parentheses fraction numerator 9 x over denominator 2 end fraction close parentheses to the power of negative 1 end exponent

straight g left parenthesis x right parenthesis space equals space square root of 1 space minus space p x cubed end root where space p is a constant.

Given that straight f open parentheses 1 third close parentheses equals straight g open parentheses 1 third close parentheses work out the value of space p.

How did you do?

2a
Sme Calculator1 mark

straight f left parenthesis x right parenthesis equals left parenthesis x plus 2 right parenthesis cubed

straight g is a function such that  gf left parenthesis x right parenthesis equals left parenthesis x plus 2 right parenthesis to the power of 12

Work out an expression for straight g left parenthesis x right parenthesis.

How did you do?

2b
Sme Calculator2 marks

straight h left parenthesis x right parenthesis equals x squared plus 5

straight k is a function such that hk left parenthesis x right parenthesis equals 4 x squared plus 5

Work out an expression for  kh left parenthesis x right parenthesis.

How did you do?

3
Sme Calculator4 marks

straight f open parentheses x close parentheses is a function with domain all values of x.

straight f left parenthesis x right parenthesis equals square root of x squared plus 6 x minus a end root  where space a spaceis a constant.

Work out the possible values of a. Give your answer as an inequality.

How did you do?

4
Sme Calculator6 marks

For each of these two function machines, when the input is a the output is b.

k greater than 0    and    k not equal to 1    and     a greater than 0

Two flowcharts: Top processes 'a' as square, add 3, multiply by k, yielding 'b'. Bottom processes 'a' as multiply by k, square, add 3, yielding 'b'.

Work out an expression for a in terms of k.

Give your answer in its simplest form.

How did you do?