The values of two functions, f and g, are given in a table. One, both, or neither of...

The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Decide which, if any, are exponential, and give the exponential models for those that are. HINT [See Example 1.]

Example 1:

Some of the values of two functions, f and g, are given in the following table:

One of these functions is linear, and the other is exponential. Which is which?

Solution Remember that a linear function increases (or decreases) by the same amount every time x increases by 1. The values of f behave this way: Every time x increases by 1, the value of f (x) increases by 4. Therefore, f is a linear function with a slope of 4. Because f (0) = 1, we see that

f (x) = 4x + 1

is a linear formula that fits the data.

On the other hand, every time x increases by 1, the value of g(x) is multiplied by 3. Because g(0) = 2, we find that

is an exponential function fitting the data.

We can visualize the two functions f and g by plotting the data points (Figure 11). The data points for f (x) clearly lie along a straight line, whereas the points for g(x) lie along a curve. The y coordinate of each point for g(x) is 3 times the y coordinate of the preceding point, demonstrating that the curve is an exponential one.