Question
Some values of the linear function f are shown in the table above. Which of the following defines f ?
- f(x) =2x + 3
- f(x) =3x + 2
- f(x) =4x + 1
- f(x) = 5x
Hint:
Hint:
Linear functions are those whose graph is a straight line. A linear function has the following form. y = f(x) = mx + c. A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.
The correct answer is: f(x) =4x + 1
Solution:
It is given that f is a linear function of x, so the standard linear equation
f(x) = mx + c,
where m and c are constants,
This equation can be used to define the relationship between x and f(x).
In this equation, m represents the increase in the value of f(x) for every increase in the value of x by 1.
From the table, it can be determined that the value of f(x) increases by 8 for every increase in the value of x by 2.
i.e. for x = 1 , f(x) = 5
x= 2 , f(x) = 13
Similarly , in other words, for the function f the value of m is 
, or 4.
The value of b can be found by substituting the values of x and f(x) from any row of the table and the value of m into the equation f(x) = mx + b and solving for b.
Using x = 1, f(x) = 5, and m = 4
5 = 4(1) + b.
Subtract 4 from both sides, we get
b = 1
Putting the values of m and b in the equation we get,
f(x) = 4x + 1
Therefore, the equation defining the function f can be written in the form
f(x) = 4x + 1.
Any equation defining the linear function f must give values of f(x) for corresponding values of x, as shown in each row of the table.
According to the table, if x = 3, f(x) = 13
For option A , substituting x = 3 gives
f(3) = 2(3) + 3, or f(3) = 9, not 13.
Similarly, for option B substituting x = 3 into the equation gives
f(3) = 3(3) + 2, or f(3) = 11, not 13.
Lastly, for option D substituting x = 3 into the equation gives
f(3) = 5(3), or f(3) = 15, not 13.
Therefore, the equations in choices A, B, and D cannot define f. Options A, B, and D are incorrect.
Therefore, the correct option is C) f(x) = 4x + 1.
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