DrDelMath

Special Topics

 

What To Do When Confronted By A Function

Before explaining how to react to a function, it must be emphasized that you do not solve a function.

When confronted with a function you should make note of and write down everything you know about that function and can deduce about that function.

Here is a list of some of the questions you should try to answer about the function. Assume its name is f.

1. What is its domain ?

2. What is its range ?

3. What kind of function is it (linear, quadratic, general polynomial of degree?, trignometric, exponential, rational, etc.) ?

4. What is the shape of its graph ?

5. How many zeros do you expect ?

6. What are its zeros? Where is f(x) = 0 ?

7. What are its real zeros ?

8. Where does it cross the x-axis ?

9. Where is it positive ? Where is f(x) > 0 ?

10. Where is it negative ? Where is f(x) < 0 ?

11. Where is its graph above the x-axis ?

12. Where is its graph below the x-axis ?

14. Where is it increasing ?

15. Where is it decreasing ?

16. Is it continuous ?

17. What are its points of discontinuity ?

18. What is its behavior as x approaches infinity ?

19. What is its behavior as x approaches negative infinity ?

20. Does it have any horizontal tangents ? Where are they?

21. What are its relative maxima ?

22. What are its relative minima ?

23. What are its absolute maxima ?

24. What are its absolute minima ?

25. Where is it increasing ?

26. Where is it decreasing ?

27. Where is it concave up ?

28. Where is it concave down ?



A TYPICAL KIND OF QUESTION

Many questions you ask yourself when analyzing a function (and others ask about a function) inquire where a function exhibits certain behaviors.

For example: Where does the graph cross the x-axis ?
Where is the graph above the x axis ?
Where is f(x) < 0 ?
Where does the graph of a function f intersect the graph of another function g ?

In almost every case:
The question is about behavior in the range and the answer is a subset of the domain.

Example 1:
The question where is f(x) > 0 asks where are range elements positive

An acceptable answer to this question will specify a subset A of the domain D with the property that every element of that subset A is associated with a positive number in the range.
Moreover, no element of the domain D outside the subset A is associated with a positive element of the range.

Example 2:
The question where is f(x) = 0 asks where are range elements 0

An acceptable answer to this question will specify a subset K of the domain D with the property that every element of that subset K is associated with the number 0 in the range.
Moreover, no element of the domain D outside the subset K is associated with the number 0 in the range.
Notice that each element of this set K is a zero of the function.

Example 3:
Where does the graph of a function f cross the horizontal line y = 8 ?
Clearly it crosses that line when its second coordinate is 8.
So the question really is Where is f(x) = 8 which asks where are range elements equal to 8 ?

An acceptable answer to this question will specify a subset J of the domain D with the property that every element of that subset J is associated with the number 8 in the range.
Moreover, no element of the domain D outside the subset J is associated with the number 8 in the range.

Example 4:
Where do the graphs of two functions with a common domain f and g intersect? This really is the question; where are the range elements f(x) and g(x) equal. That is, for what domain elements x is it true that f(x) = g(x).

An acceptable answer to this question will specify a subset H of the common domain D with the property that for each individual element h of H it is true that g(h) = f(h)
Moreover, for every element m not in the set H, g(m) is not equal to f(m)

SOME SPECIFIC EXAMPLES

Example 1:
Consider the function whose rule is f(x) = 2x + 5. That is try to answer as many of the preceding questions as you can. For now restrict your attention to the first 15 questions.

1. What is its domain ? Answer: By convention the domain of f is R

2. What is its range ? Answer: By convention the range of f is R

3. What kind of function is it ? (linear, quadratic, general polynomial of degree, trignometric, exponential, rational, etc.)Answer: The function f is a linear function.

4. What is the shape of its graph ? Answer: Because f is a linear function its graph will be a non-vertical line. The possibilities look like one of the following:      

Because the leading coefficient is positive, the graph must look like the following:
(This last observation can only be made for linear functions.)    

5. How many zeros do you expect ? Answer: Linear functions have no zeros or one zero.

6. What are its zeros? Where is f(x) = 0 ? Answer: To find the zeros of f we must solve the equation resulting from f(x) = 0. In this case we must solve the equation 2x + 5 = 0. Its solution set is { - 5/2 }. Therefore f has one zero and it is - 5/2.

7. What are its real zeros ? Answer: The one zero of f is a real number, so the real zero of f is - 5/2.

8. Where does it cross the x-axis ? Answer: The graph of any function crosses the x-axis at the real zeros of the function. Therefore, in this case, the graph of the function f crosses the x-axis at - 5/2. This gives us more information about the graph of the function f.                 

11. Where is its graph above the x-axis ? Answer: The graph of f is above the x-axis for all domain elements in the set { x | x R and x > -5/2 }.

12. Where is its graph below the x-axis ? Answer: The graph of f is below the x-axis for all domain elements in the set { x | x R and x < -5/2 }.

9. Where is it positive ? Where is f(x) > 0 ? Answer: A function is said to be positive at a domain element t if f(t) > 0 because the corresponding range element is positive. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is positive, the point (t, f(t)) must appear above the x-axis. Therefore the function f is positive for all domain elements where the graph appears above the x-axis. In this case f is positive in the set { x | x R and x > -5/2 }.

10. Where is it negative ? Where is f(x) < 0 ? Answer: A function is said to be negative at a domain element t if f(t) < 0 because the corresponding range element is negative. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is negative, the point (t, f(t)) must appear below the x-axis. Therefore the function f is negative for all domain elements where the graph appears below the x-axis. In this case f is negative in the set { x | x R and x < -5/2 }.

14. Where is it increasing ? Answer: As you move from left to right in the domain, the functional values increase ( w < h ? f(w) < f(h) for all real numbers. ) Therefore f is increasing for all real numbers.

15. Where is it decreasing ? Answer: Since f is increasing for all real numbers it is nowhere decreasing.

Example 2:
Consider the function whose rule is f(x) = -2x + 5. That is try to answer as many of the preceding questions as you can. For now restrict your attention to the first 15 questions.

1. What is its domain ? Answer: By convention the domain of f is R

2. What is its range ?
Answer: By convention the range of f is R

3. What kind of function is it ? (linear, quadratic, general polynomial of degree, trignometric, exponential, rational, etc.)
Answer: The function f is a linear function.

4. What is the shape of its graph ?
Answer: Because f is a linear function its graph will be a non-vertical line. The possibilities look like one of the following:   


Because the leading coefficient is negative, the graph must look like the following:
(This last observation can only be made for linear functions.)   


5. How many zeros do you expect ?
Answer: Linear functions have no zeros or one zero.

6. What are its zeros? Where is f(x) = 0 ?
Answer: To find the zeros of f we must solve the equation resulting from f(x) = 0. In this case we must solve the equation -2x + 5 = 0. Its solution set is { 5/2 }. Therefore f has one zero and it is 5/2.

7. What are its real zeros ?
Answer: The one zero of f is a real number, so the real zero of f is 5/2.

8. Where does it cross the x-axis ?
Answer: The graph of any function crosses the x-axis at the real zeros of the function. Therefore, in this case, the graph of the function f crosses the x-axis at 5/2. This gives us more information about the graph of the function f.                

11. Where is its graph above the x-axis ? Answer: The graph of f is above the x-axis for all domain elements in the set { x | x R and x < 5/2 }.

12. Where is its graph below the x-axis ?
Answer: The graph of f is below the x-axis for all domain elements in the set { x | xR and x > 5/2 }.

9. Where is it positive ? Where is f(x) > 0 ?
Answer: A function is said to be positive at a domain element t if f(t) > 0 because the corresponding range element is positive. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is positive, the point (t, f(t)) must appear above the x-axis. Therefore the function f is positive for all domain elements where the graph appears above the x-axis. In this case f is positive in the set { x | x R and x < 5/2 }.

10. Where is it negative ? Where is f(x) < 0 ? A function is said to be negative at a domain element t if f(t) < 0 because the corresponding range element is negative. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is negative, the point (t, f(t)) must appear below the x-axis. Therefore the function f is negative for all domain elements where the graph appears below the x-axis. In this case f is negative in the set { x | x R and x > 5/2 }.

15. Where is it decreasing ? Answer: As you move from left to right in the domain, the functional values decrease ( w < h ? f(w) > f(h) for all real numbers. ) Therefore f is decreasing for all real numbers.

14. Where is it increasing ? Answer: Since f is decreasing for all real numbers it is nowhere increasing.

Example 3:
Consider the function whose rule is f(x) = x2 + x - 6. That is try to answer as many of the preceding questions as you can. For now restrict your attention to the first 15 questions.

1. What is its domain ? Answer: By convention the domain of f is R

2. What is its range ? Answer: By convention the range of f is R

3. What kind of function is it ?
Answer: (linear, quadratic, general polynomial of degree, trignometric, exponential, rational, etc.) The function f is a quadratic function. Because its rule is of the form f(x) = ax2 + bx + c and a is not zero. Note a = 1, b = 1 and c = - 6
4. What is the shape of its graph ? Answer: Because f is a quadratic function its graph will be parabola which opens up or a parabola which opens down. The possibilities look like one of the following:

Because the leading coefficient is positive, the parabola opens up and the graph must look like the following:

(This last observation can only be made for quadratic functions.)
5. How many zeros do you expect ?
Answer: Quadratic functions have no zeros, one zero or two zeros.

6. What are its zeros? Where is f(x) = 0 ? Answer: To find the zeros of f we must solve the equation resulting from f(x) = 0. In this case we must solve the equation x2 + x - 6 = (x + 3)(x - 2) = 0. Its solution set is { -3, 2 }. Therefore f has two zeros and they are -3 and 2.
7. What are its real zeros ? Answer: Both zeros of f are real numbers, so the real zeros of f are - 3 and 2.

8. Where does it cross the x-axis ?
Answer: The graph of any function crosses the x-axis at the real zeros of the function. Therefore, in this case, the graph of the function f crosses the x-axis at - 3 and 2. This gives us more information about the graph of the function f.        

11. Where is its graph above the x-axis ? Answer: The graph of f is above the x-axis for all domain elements in the set { x R | x < -3 or x > 2 } = ( - , -3 ) ( 2, + )

12. Where is its graph below the x-axis ?
Answer: The graph of f is below the x-axis for all domain elements in the set { x R | x > -3 and x < 2 } = { x R | -3 < x < 2 } = ( -3, 2 )

9. Where is it positive ? Where is f(x) > 0 ?
Answer: A function is said to be positive at a domain element t if f(t) > 0 because the corresponding range element is positive. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is positive, the point (t, f(t)) must appear above the x-axis. Therefore the function f is positive for all domain elements where the graph appears above the x-axis. In this case f is positive in the set { x R | x < -3 or x > 2 }.

10. Where is it negative ? Where is f(x) < 0 ? Answer: A function is said to be negative at a domain element t if f(t) < 0 because the corresponding range element is negative. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is negative, the point (t, f(t)) must appear below the x-axis. Therefore the function f is negative for all domain elements where the graph appears below the x-axis. In this case f is negative in the set { x R | -3 < x < 2 }.

15. Where is it decreasing ?
Answer: As you move from left to right in the domain, the functional values decrease ( w < h f(w) > f(h) ) for all real numbers to the left of the vertex. Therefore f is decreasing for all real numbers to the left of the vertex. The first coordinate of the vertex of a quadratic function is at x = -b/2a. So the first coordinate of the vertex of this function f is x = -1/2. Therefore the function is decreasing for all x in the set { x R | x < - 1/2} = ( - , -1/2 )

14. Where is it increasing ? Answer: As you move from left to right in the domain, the functional values increase ( w < h f(w) < f(h) ) for all real numbers to the right of the vertex. Therefore f is increasing for all real numbers to the right of the vertex. The first coordinate of the vertex of a quadratic function is at x = -b/2a. So the first coordinate of the vertex of this function f is x = -1/2. Therefore the function is increasing for all x in the set { x R | x > - 1/2} = ( -1/2 , ).

The vertex of the graph of f is a point on the graph of f and therefore its coordinates must satisfy the rule for the function f.
Note that the vertex of this function f is therefore at ( -1/2, f(-1/2) ).
But f(-1/2) = (-1/2)2 + (-1/2) - 6 = 1/4 - 1/2 - 6 = (1 -2 -24)/4 = -25/4
So the vertex of this parabola is the point ( -1/2, -25/4 ).

Example 4:
Consider the function whose rule is f(x) = -x2 - 3x + 4. That is try to answer as many of the preceding questions as you can. For now restrict your attention to the first 15 questions.

1. What is its domain ? Answer: By convention the domain of f is R

2. What is its range ?
Answer: By convention the range of f is R

3. What kind of function is it ? Answer: (linear, quadratic, general polynomial of degree, trignometric, exponential, rational, etc.) The function f is a quadratic function. Because its rule is of the form f(x) = ax2 + bx + c and a is not zero. Note a = -1, b = -3 and c = 4

4. What is the shape of its graph ? Answer: Because f is a quadratic function its graph will be parabola which opens up or a parabola which opens down. The possibilities look like one of the following:

Because the leading coefficient is negative, the parabola opens down and the graph must look like the following:
(This last observation can only be made for quadratic functions.)

5. How many zeros do you expect ? Answer: Quadratic functions have no zeros, one zero or two zeros.

6. What are its zeros? Where is f(x) = 0 ? Answer: To find the zeros of f we must solve the equation resulting from f(x) = 0. In this case we must solve the equation -x2 - 3x + 4 = (-x + 1)(x + 4) = 0. Its solution set is { -4, 1 }. Therefore f has two zeros and they are -4 and 1.

7. What are its real zeros ? Answer: Both zeros of f are real numbers, so the real zeros of f are - 4 and 1.

8. Where does it cross the x-axis ? Answer: The graph of any function crosses the x-axis at the real zeros of the function. Therefore, in this case, the graph of the function f crosses the x-axis at -4 and 1. This gives us more information about the graph of the function f.    

11. Where is its graph above the x-axis ? Answer: The graph of f is above the x-axis for all domain elements in the set { x R | -4 < x < 1 } = ( -4, 1 )

12. Where is its graph below the x-axis ? Answer: The graph of f is below the x-axis for all domain elements in the set { x R | x < -4 or x > 1 } = ( - , -4 )( 1, + )

9. Where is it positive ? Where is f(x) > 0 ? Answer: A function is said to be positive at a domain element t if f(t) > 0 because the corresponding range element is positive. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is positive, the point (t, f(t)) must appear above the x-axis. Therefore the function f is positive for all domain elements where the graph appears above the x-axis. In this case f is positive in the set ( -4, 1 ).

10. Where is it negative ? Where is f(x) < 0 ?
Answer: A function is said to be negative at a domain element t if f(t) < 0 because the corresponding range element is negative. Recall that for any domain element t, the point (t, f(t)) is on the graph of the function f. If f(t) is negative, the point (t, f(t)) must appear below the x-axis. Therefore the function f is negative for all domain elements where the graph appears below the x-axis. In this case f is negative in the set ( - , -4 )( 1, + ).

15. Where is it decreasing ? Answer: As you move from left to right in the domain, the functional values decrease ( w < h f(w) > f(h) ) for all real numbers to the right of the vertex. Therefore f is decreasing for all real numbers to the right of the vertex. The first coordinate of the vertex of a quadratic function is at x = -b/2a. So the first coordinate of the vertex of this function f is x = -3/2. Therefore the function is decreasing for all x in the set { x R | x > - 3/2} = ( -3/2, - )

14. Where is it increasing ?
Answer: As you move from left to right in the domain, the functional values increase ( w < h f(w) < f(h) ) for all real numbers to the left of the vertex. Therefore f is increasing for all real numbers to the left of the vertex. The first coordinate of the vertex of a quadratic function is at x = -b/2a. So the first coordinate of the vertex of this function f is x = -3/2. Therefore the function is increasing for all x in the set { x R | x < - 3/2} = ( - , -3/2).

The vertex of the graph of f is a point on the graph of f and therefore its coordinates must satisfy the rule for the function f.
Note that the vertex of this function f is therefore at ( -3/2, f(-3/2) ).
But f(-3/2) = - (-3/2)2 - 3(-3/2) + 4 = - 9/4 + 9/2 + 4 = (-9 + 18 + 16)/4 = 25/4
So the vertex of this parabola is the point ( -3/2, 25/4 ).