The table shows the outputs, y, for different inputs, x:
Input (x)
13
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)
There is one and only one value of y for each value of x, so yes
4(13) + 8 = 60
4(5) + 8 = 28
4(7)+8 = 36
or
4(3)+8 = 20
4 x + 8 = 76
4 x = 68
x = 17
Part A: To determine if the data in the table represent a function, we can check if each input value (x) is associated with a unique output value (y). In this table, we can see that each x-value has only one corresponding y-value, which means there is no repetition in the x-values. Therefore, the table does represent a function.
Part B: To compare the data in the table with the relation f(x) = 4x + 8, we can substitute x = 3 into both the table and the equation.
In the table, when x = 3, the corresponding value of y is 5.
Using the equation f(x) = 4x + 8, when x = 3:
f(x) = 4(3) + 8
f(x) = 12 + 8
f(x) = 20
Therefore, the value of y in the table (5) is smaller than the value of f(x) in the relation f(x) = 4x + 8 (20) when x = 3.
Part C: To find the value of x when f(x) = 76, we can set up the equation f(x) = 76 and solve for x.
Using the equation f(x) = 4x + 8, we replace f(x) with 76:
76 = 4x + 8
To isolate the variable x, we can subtract 8 from both sides of the equation:
76 - 8 = 4x
68 = 4x
Lastly, we divide both sides of the equation by 4 to solve for x:
68 / 4 = x
17 = x
Therefore, the value of x for which f(x) = 76 is x = 17.