x f(x) g(x) ∆x ∆f ∆g ∆f/∆x ∆g/∆x
0 0 -1
3 3 27 1 9
3 3 26
3 44 27 44/3 9
6 47 53
4 30 36 15/2 9
10 77 89
5 37 45 37/5 9
15 114 134
The table shows the value x and the corresponding values of f(x) and g(x). The ∆x column shows the difference between the x values from the preceding row and the next row. ∆f and ∆g columns show the corresponding differences for each of the two functions. A linear function has a constant gradient, and the gradients for f and g are given by the division ∆f/∆x and ∆g/∆x in the last two columns. We can see that g(x) has a constant gradient 9, so it must be a linear function. ∆f/∆x is clearly not linear because the gradients are not constant.
If we choose a point (0,-1) we can write the equation knowing the gradient: g(x)-(-1)=9x. So g(x)+1=9x and g(x)=9x-1.