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mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Introduction to Functions

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:introduction_to_functions/4620/


Example 1.

Construct and fill in the table for the function . For the variable , choose the numbers from to with a step of .

    
Example 2.

A description is given of how the function f (x) changes the variable x. Write down the function rule for the following:


  • Function f (x) reduces the number x by .

  • A function reduces the number x by .

  • The dependent variable is twice the independent variable in a function.

  • The dependent variable is times less than twice the independent variable in a function.

  • The value f (x) is less than a quarter of the number x.

  • The dependent variable is greater than five times the independent value in a function.

  • The function f (x) is greater than the additive inverse of the number x.

  • The dependent variable is less than six times the additive inverse of the independent variable x.

  • A function decreases the multiplicative inverse of the number x by .

  • The dependent variable of a function increases the multiplicative inverse of the number x by .

  • The function f is the square of the independent variable.

  • A function increases the absolute value of the number x by .

  • The dependent variable of a function is the absolute value of the multiplicative inverse of the independent variable.

  • A function is one-fifth of the sum of the numbers x and .

  • The dependent variable is half the difference of the additive inverse of the number x and .


    
3.

Calculate the value of the statement:




if statements are made






    
4.

Calculate the value of the function for:


  • ,


  • .

    
5.

We want to calculate the perimeter of a rhombus.

  • What does the perimeter of a rhombus depend on?

  • What is dependent and what is an independent variable?

  • Write down the rule that determines the dependency.

    
6.

Let's have a function . The table shows the values of the function.


xf (x) 
-53 
-2-7 
-113 
3-6 
44 
745 

Write down the following values of the function:








    
7.

From the graph of the function f read the following values:











    
8.

Let's have a function f, with the rule .


  • Complete the function table:


    xf (x) 
    4  
      
    -2  
      
    12  
    6  
    -5  


  • The figure shows the graph of the function f:




    Read the following values from the graph:






    

9.

The function f with the regulation is given. The values f (x) are written in the table. Calculate the values x and fill in the table.


xf (x) 
 8 
 2 
 17 
 -13 
 5 
 23 
 -10 
 -4 
    
10.

Let's have a function f with the rule. Use the arrow diagram to show how the function maps integers from to .

    
11.

From the given arrow diagrams, write the rule of the function according to which the independent value is mapped.

Arrow diagrams

    
12.

Let us have the function , with the rule . For each integer from to , write down where the f function maps it to.

    
13.

Write down the given function rules in words.


















    
14.

Determine the function values ​​for the function f, which is determined by the rule: f(x) is a quarter of the sum of the number x and . Determine the values ​​for the integers from to with a step of .

    
15.

Functions f and g are given:






Calculate the following function values:







    
16.

Let us have a cuboid with length , width and height . The width is equal to half the length, and the height is equal to a quarter of the length.


  • Write the expression , which represents the dependence of the volume on the length of the side ,

  • Calculate and ,

  • Write the expression , which represents the dependence of the side on the volume ,

  • Calculate and .

    
17.

Find the points on the graph of the function from the given function table and draw them in the coordinate system.


xf(x) 
03 
1 
24 
-1 
-60 
-41 
    
18.

Find the values ​​from the graph of the function and fill in the table below. Draw an arrow diagram from the values ​​in the table.




xf(x) 
5  
-2  
  
 -2 
4  
    

19.

Read the points from the given arrow diagram and draw them in the coordinate system.


Arrow diagram

    
20.

A function is given by:




Plot the points of the function from the given expression and draw the graph of the function in the coordinate system.


For values x, take the set .

    
21.

Does the drawn curve represent the graph of a function?


    
22.

A cyclist riding towards us at a constant speed was away from us when we started measuring time.


  • Write down the distance of the rider from us as a function of time.


  • How many meters is the cyclist away from us after 6 seconds?


  • When will the cyclist pass us?


  • Graphically represent the distance of the cyclist from us as a function of time.

    
23.

Write down the zeros of the real function whose graph is in the figure.


    
24.

The graph of the function f(x) is drawn below. Write down the zeros of the function and write down the point at which the graph of the function f(x) intersects the ordinate axis.


    
25.

A real function is given. Write its zero, the intersection with the ordinate axis, and draw its graph .

    
26.

A real function is given. Write its zero, the intersection with the ordinate axis, and draw its graph.

    
27.

The real function (), tabulate on the interval [-3, 3] with step 1. Write down the zeros of the function and the intersection of the graph with the ordinate axis .

    
28.

Draw a graph of the function:




for and write down its range.

    

29.

A real function is given.


  • Solve the equation for by factorizing.


  • Make a table of value for the function on the interval [-1, 6] with step 1 and write the zeros and the intersection of the graph with the ordinate axis.


  • Draw a graph of the function f.

    
30.

Make a table of the function , on the interval [-3, 3] with step 1.


  • Write down the zeros of the function and the intersection of the graph with the ordinate axis.


  • Draw a graph of the function f.

    
31.

The function f is given by an arrow diagram. Justify whether the function is injective, surjective, or bijective.


    
32.

The function f is given by an arrow diagram. Justify whether the function is injective, surjective, or bijective.


    
33.

The function f is given by an arrow diagram. Justify whether the function is injective, surjective, or bijective.


    
34.

The function f is given by an arrow diagram. Justify whether the function is injective, surjective, or bijective.


    
35.

A graph of a real function f is given. Justify whether the function is injective, surjective, or bijective.


    
36.

Draw a graph of the real function and write down its range.

    
37.

The function is given.


  • Specifies the zero and initial value of the function.


  • Draw a graph of the function.


  • Specifies the domain and range of the function .

    
38.

We have a function . Calculate the numbers a and b so that the function belongs to the graph below.