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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.

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