For each function listed below evaluate the function at each specified value of the independent variable. These instructions are more appropriately stated:

For each function listed below compute the unique range element associated with the specified domain element

23) The rule for a function named f is given by the equation  f(x) = 2x - 3.
Find f(1). That is, compute the unique range element associated with the domain element 1.
According to the rule for f, f(1) = 2(1) - 3 = -1
-1 is the unique range element associated with the domain element 1 by the function named f.
The following observations should also be made.
The coordinates of the point (1, -1) satisfy the rule of the function.
The point (1, -1) is on the graph of the function f.

Find f(-3). That is, compute the unique range element associated with the domain element -3.
According to the rule for f, f(-3) = 2(-3) -3 = -9
-9 is the unique range element associated with the domain element -3 by the function named f.
The following observations should also be made.
The coordinates of the point (-3, -9) satisfy the rule of the function.
The point (-3, -9) is on the graph of the function f.

Find f(x-1). That is, compute the unique range element associated with the domain element (x - 1).
According to the rule for f, f(x - 1) = 2(x - 1) -3 = 2x -2 - 3 = 2x -5

24)  The rule for a function named g is given by the equation g(y) = 7 - 3y
Do not confuse the variable y used here with the "y-axis". As used here, y is simply a variable which represents an arbitraty domain element.
Find g(0). That is, find the unique element of the range with is associated with the domain element 0 by the function g.
According to the rule for g, g(0) = 7 - 3(0) = 7
7 is the unique range element associated with the domain element 0 by the function named f.
The following observations should also be made.
The coordinates of the point (0, 7) satisfy the rule of the function.
The point (0, 7) is on the graph of the function f, in fact it is the y-intercept of the graph.

Find g(s + 2). That is, find the unique element of the range with is associated with the domain element (s - 2)by the function g.
According to the rule for g, g(s - 2) = 7 - 3(s - 2) = 7 - 3s + 6 = 13 - 3s

43) The rule for a function named f is f(x) = 15 - 3x.
      Find all real values of x for which f(x) = 0.
Solution: The statement of the question implies 0 = f(x) = 15 - 3x.
We are therefore asked to find the real solutions of the equation 0 = 15 - 3x
The normal methods of solving a linear equation yields:
0 = 15 - 3x
3x = 15
x = 5
Conclusion: 5 is the only real number for which f(x) = 0
Another way of stating this is: 5 is the only real zero of the function f.
This implies that the graph of the function f intersects the x-axis at 5.
The x-intercept of the graph of the function f is (5, 0).

45) The rule for a function named f is .
      Find all real values of x for which f(x) = 0.
Solution: The statement of the question implies 0 = .
We are therefore asked to find the real solutions of the equation
The normal methods of solving a linear equation yields:

0 = 3x - 4
3x = 4
x =
Conclusion: is the only real number for which f(x) = 0
Another way of stating this is: is the only real zero of the function f.
This implies that the graph of the function f intersects the x-axis at .
The x-intercept of the graph of the function f is .

48) The rule for a function named f is f(x) = x² - 8x + 15.
      Find all real values of x for which f(x) = 0.
Solution: The statement of the question implies 0 = f(x) = x² - 8x + 15.
We are therefore asked to find the real solutions of the equation 0 = x² - 8x + 15
The normal methods of solving a quadratic (factoring in this case) equation yields:
x² - 8x + 15 = 0
(x - 5)(x - 3) = 0
x - 5 = 0   or   x - 3 = 0
x = 5   or   x = 3

Conclusion: 5 and 3 are the two real numbers for which f(x) = 0
Another way of stating this is: 5 and 3 are the two real zeros of the function f.
This implies that the graph of the function f intersects the x-axis at 5 and 3.
The x-intercepts of the graph of the function f are (5, 0) and (3, 0).

50) The rule for a function named f is f(x) = x³ - x² - 4x + 4.
      Find all real values of x for which f(x) = 0.
Solution: The statement of the question implies 0 = f(x) = x³ - x² - 4x + 4.
We are therefore asked to find the real solutions of the equation 0 = x³ - x² - 4x + 4
The normal methods of solving a quadratic (factoring in this case) equation yields:
x³ - x² - 4x + 4 = 0
x²(x - 1) - 4(x - 1) = 0
(x - 1)(x² - 4) = 0
(x - 1)(x - 2)(x + 2) = 0
x - 1 = 0   or   x - 2 = 0   or   x + 2 = 0
x = 1  or   x = 2   or   x = -2

Conclusion: 1, 2 and -2 are the three real numbers for which f(x) = 0
Another way of stating this is: 1, 2 and -2 are the two real zeros of the function f.
This implies that the graph of the function f intersects the x-axis at 1, 2 and -2.
The x-intercepts of the graph of the function f are (1, 0), (2, 0), and (-2, 0).

Do not be concerned with how the following graphs were obtained.
Use the picture to better understand the solution to the above question.
              

52)  If f(x) = x⁴ - 2x² and g(x) = 2x² are the rules for two functions, where is f(x) = g(x)?
Solution:  Note that the question is about equality of range elements and the answer will be elements of the domain.
The question can be rephrased by considering the extremes of    x⁴ - 2x² = f(x) = g(x) = 2x²
to get the equation
x⁴ - 2x² = 2x²
x⁴ - 4x² = 0
x²(x² - 4) = 0
x²(x - 2)(x + 2) = 0
x² = 0   or   x - 2 = 0   or x + 2 = 0
x = 0   or   x = 2   or   x = -2
Therefore f(x) = g(x) at x = 0, 2, and -2.
Therefore f(0) = g(0) and f(2) = g(2) and f(-2) = g(-2)

This tells us that the graphs of the function f and the function g intersect at the points
      (0, f(0)),   (2, f(2)),   and (-2, f(-2))
These are the same as the points
       (0, g(0)),   (2, g(2)),   and (-2, g(-2))

The rule for either function may be used to calculate the second coordinate of the points of intesection of the two graphs.
Those points are (0, 0),   (2, 8),   and (-2, 8).
Because the solution set is {0, 2, -2}, there are no other points of intersection.

Do not be concerned with how the following graphs were obtained.
Use the picture to better understand the solution to the above question.

54)  If and g(x) = 2 - x are the rules for two functions, where is f(x) = g(x)?
Solution:  Note that the question is about equality of range elements and the answer will be elements of the domain.
The question can be rephrased by considering the extremes of     = f(x) = g(x) = 2 - x
to get the equation

= 2 - x = 6 - x x = 36 -12x + x2 (not equivalent to the original equation) x2 - 13x + 36 = 0 (x - 9)(x - 4) = 0 x - 9 = 0   or   x - 4 = 0 x = 9   or   x = 4

Test 9: Test 4:

Therefore f(x) = g(x) at x = 4.
Therefore f(4) = g(4)

This tells us that the graphs of the function f and the function g intersect at the points
      (4, f(4))
These are the same as the points
       (4, g(4))

The rule for either function may be used to calculate the second coordinate of the points of intesection of the two graphs.
Those points are (4, -2).
Because the solution set is {4}, there are no other points of intersection.

Do not be concerned with how the following graphs were obtained.
Use the picture to better understand the solution to the above question.