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Solved problems for the chapter:

Graph of a Function

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Example 1.

Draw a graph of the function:

where

Example 2.

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

Draw a table of the function , on a range of [-3, 3] with each interval of and draw its graph. Explain whether the function is injective, surjective, or bijective .

Make a table of values for the real function from -3 to 3 with step 1, draw its graph and write down its domain and range.

Function f assigns to each real number from -5 to 5 its additive inverse.

Write down the expression for the function f.
Calculate
Draw its graph.

Does the drawn curve represent the graph of a function?

Tabulate the function , function on the interval [-3, 3] with step 1 and draw its graph. Explain whether the function is injective, surjective or bijective.

Draw the following functions in the same coordinate system:

10.

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

11.

The function is given:

Draw a graph of the function.
Find the domain and range.
Is the function injectable, surjective, or bijective?

12.

Draw the following functions in the same coordinate system:

13.

Draw the following functions in the same coordinate system:

14.

An arrow diagram of the function is given.

Can we assign its inverse function to the function?

15.

Draw a graph of the function .

16.

Let be a function with the domain and the range . Let be its inverse function, with the domain and the range . Explain why the following equations apply:

17.

Graphically solve the equation:

Finally, do a test and make sure the solution is correct .

18.

The function is given. Is the function its inverse function?

19.

Find the inverse of the function . Finally, test to see if the solution is correct.

20.

An arrow diagram of the function is given. Can you assign her its inverse function ?

21.

An arrow diagram of the function is given.

Does the function has an inverse function ? Can you determine the inverse of the restricted function ?

22.

Find the inverse of the function and use a test to verify that the resulting solution is correct. Also write the domain of the inverse function.

23.

Find the inverse of the function. Finally, test to see if the solution is correct.

24.

Find the inverse of the function and use a test to check if the obtained solution is correct.

25.

We have the following functions:

Specify the domain of the functions f and g.
Draw graphs of functions f and g in the same coordinate system.
Determine the oddness / eveness of the functions f and g.

26.

Draw the following functions in the same coordinate system:

27.

The function , is given.

Write down the range of the function.
Find the inverse function
Write the domain of the inverse function, . What is the relationship with the range of the function ?

28.

For each of the following statements, determine whether or not it is true and substantiate its finding.

Mapping a graph of a function over the bisector of odd quadrants gives us a graph of its inverse function.
Each function has an inverse function.
If the function is not bijective, it has no inverse function.