Does the table represent an exponential function?
Х f(x)
-1 4/5
0 4
1 20
2 100
yes - just keep multiplying by 5
To determine if the table represents an exponential function, we need to check if the ratio of the output values (f(x)) for consecutive input values (x) remains the same.
Looking at the table, let's calculate the ratios between consecutive output values:
Ratio between f(-1) and f(0) = (4/5) / 4 = 1/5
Ratio between f(0) and f(1) = 4 / 20 = 1/5
Ratio between f(1) and f(2) = 20 / 100 = 1/5
Since the ratios are consistent at 1/5, the table represents an exponential function.
In general, an exponential function is of the form: f(x) = a * (b^x), where a is a constant and b is the base of the exponential function. In this case, the constant a and the base b can be determined as follows:
To find the constant a:
We can choose any of the output values (f(x)) and divide it by the corresponding value of the base (b^x). Let's choose f(0) = 4 and x = 0:
4 = a * (b^0) = a * 1
So, a = 4.
To find the base b:
We can choose any two pairs of input and output values (x, f(x)) and create an equation using the exponential function formula. Let's use (0, 4) and (1, 20):
4 = 4 * (b^0) = 4 * 1 = 4
20 = 4 * (b^1) = 4 * b
Dividing the second equation by the first equation, we get:
5 = b
So, the base (b) of the exponential function is 5.
Therefore, the exponential function that represents the given table is f(x) = 4 * (5^x).