The table shows the outputs, y, for different inputs, x:
Input (x)
1
3
5
7
Output (y)
8
6
5
4
Part A: Do the data in this table represent a function? Justify your answer. (3 points)
Part B: Compare the data in the table with the relation f(x) = 4x + 8. Which relation has a greater value when x = 3? (2 points)
Part C: Using the relation in Part B, what is the value of x if f(x) = 76? (5 points)
looks like for every value of x there is only one value of y, so yes, it is a function.
b) well, just evaluate f(3)
c) set 4x+8 = 76 and solve for x
Part A: To determine if the table represents a function, we need to check if each input value, x, is associated with only one output value, y. Looking at the table, each input value, x, is indeed associated with only one output value, y. Therefore, the data in this table represents a function.
Part B: Let's compare the data in the table with the relation f(x) = 4x + 8. When x = 3, substituting it into the relation gives:
f(3) = 4(3) + 8
f(3) = 12 + 8
f(3) = 20
The relation f(x) = 4x + 8 has a value of 20 when x = 3.
Part C: Now, using the relation in Part B (f(x) = 4x + 8), we need to find the value of x when f(x) = 76. To do this, we can set up the equation:
76 = 4x + 8
Subtracting 8 from both sides:
68 = 4x
Dividing both sides by 4:
x = 68 / 4
x = 17
Therefore, when f(x) = 76, the value of x is 17.
Part A: To determine if the data in the table represents a function, we need to check if each input has a unique corresponding output. In a function, every input can only have one output.
Looking at the table, we can see that each input value has only one corresponding output value. For example, when x is 1, the output is 8. When x is 3, the output is 6. There are no repeating x-values with different y-values.
Therefore, based on the given table, the data represents a function.
Part B: To compare the data in the table with the relation f(x) = 4x + 8, we need to substitute x = 3 into both expressions and compare the resulting values.
For the given data in the table, when x = 3, the output value (y) is 6.
Let's substitute x = 3 into the relation f(x) = 4x + 8:
f(3) = 4(3) + 8 = 12 + 8 = 20
The relation f(x) = 4x + 8 has a greater value when x = 3, as it yields f(3) = 20, which is greater than the output value 6 in the table.
Part C: To find the value of x when f(x) = 76 using the relation f(x) = 4x + 8, we need to solve the equation:
4x + 8 = 76
First, we subtract 8 from both sides of the equation:
4x = 76 - 8
4x = 68
Then, we divide both sides by 4 to isolate x:
x = 68 / 4
x = 17
Therefore, according to the relation f(x) = 4x + 8, when f(x) = 76, the value of x is 17.