When working with functions, which involve the enhancement operation or a variable like power, they are called exponential functions. Exponential functions are generally expressed as follows:
y (x) = kax
Where, k is a real number, a is a positive number different from one and x, in addition to being the variable, is the power.
To graph an exponential function, it is important to remember the power property that says that every number raised to zero is equal to one, that is, to0 = 1. So, being the increasing or decreasing graph it will always cut to the y axis at 1.
Every function f: R → R such that y (x) = ax , where a is positive but different from one, is called the exponential function.
- Like0 = 1, the curve passes through the point (0,1).
- Like1 = a, the curve passes through point (1, a).
- The value of y in the expression y = ax for any number in the set R is always a positive number and can never be zero.
- The curve never cuts the x axis.
- When a> 1 the curve is strictly increasing.
- When at <1 the curve is strictly decreasing.
The exponential function is always positive, therefore its domain will be from (-∞, + ∞) and its range from (0, + ∞).
Example: graph the following function f (x) = 2(1-x) and determine your domain and range.
The fact that the variable in the function is found as a power, confirms that this function is exponential.
The function can be rewritten as follows:
f (x) = 2- (x-1)
Recalling the power property at-1 = 1 / a, then:
f (x) = 1/2(x-1)
We observe that the base of this exponential function is less than one, therefore its graph is decreasing. But in order to graph it, we must know at what point it intersects the axis and, for this, we will substitute the value x = 0 in the function:
f (x) = 1/2(0-1)
f (x) = 1/2-1
Applying the power property 1 / a = a-1, so:
f (x) = 2
The function cuts the y-axis at 2. Let's look at the graph:
Function domain: (-∞, + ∞)
Function range: (0, + ∞)