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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