Graph of Exponential Functions

We first start with the properties of the graph of the basic exponential function of base a,

The domain of function f is the set of all real numbers. The range of f is the interval (0 , +infinity).

The graph of f has a horizontal asymptote given by y = 0. Function f has a y intercept at (0 , 1). f is an increasing function if a is greater than 1 and a decreasing function if a is smaller than 1 .

You may want to review all the above properties of the exponential function interactively .

Example 1: f is a function given by

1. Find the domain and range of f.
   
2. Find the horizontal asymptote of the graph.
   
3. Find the x and y intercepts of the graph. of f if there are any.
   
4. Sketch the graph of f.

Answer to Example 1

1. The domain of f is the set of all real numbers. To find the range of f,we start with
   2^x > 0
   
   Multiply both sides by 2^-2 which is positive.
   2^x2^-2 > 0

   Use exponential properties
   2^{(x - 2)} > 0
   
   This last statement suggests that f(x) > 0. The range of f is (0, +inf).
   
   
2. As x decreases without bound, f(x) = 2^{(x - 2)} approaches 0. The graph of f has a horizontal asymptote at y = 0.
   
   
3. To find the x intercept we need to solve the equation
   f(x) = 0
   
   
   2^{(x - 2)}  = 0
   
   This equation does not have a solution, see range above, f(x) > 0. The graph of f does not have an x intercept. The y intercept is given by
   (0 , f(0)) = (0,2^{(0 - 2)}) = (0 , 1/4).
   
   So far we have the domain, range, y intercept and the horizontal asymptote. We need extra points.
   (4 , f(4)) = (4, 2^{(4 - 2)}) = (4 , 2²) = (4 , 4)
   
   
   (-1 , f(-2)) = (-1, 2^{(-1 - 2)}) = (-1 , 2^-3) = (-1 , 1/8)
   
   
   
4. Let us now use all the above information to graph f.
   
   

Matched Problem to Example1:  f is a function given by

1. Find the domain and range of f.
   
2. Find the horizontal asymptote of the graph of f.
   
3. Find the x and y intercepts of the graph of f if there are any.
   
4. Sketch the graph of f.

1. The domain of f is the set of all real numbers. To find the range of f, we start with
   3^x > 0
   
   Multiply both sides by 3 which is positive.
   3^x3 > 0
   
   Use exponential properties
   3^{(x + 1)} > 0
   
   Subtract 2 to both sides
   3^{(x + 1)} -2 > -2
   
   This last statement suggests that f(x) > -2. The range of f is (-2, +inf).
   
   
2. As x decreases without bound, f(x) = 3^{(x + 1)} -2 approaches -2. The graph of f has a horizontal asymptote y = -2.
   
   
3. To find the x intercept we need to solve the equation f(x) = 0
   3^{(x + 1)} - 2  = 0
   
   Add 2 to both sides of the equation
   3^{(x + 1)} = 2
   
   Rewrite the above equation in Logarithmic form
   x +1 = log₃ 2
   
   Solve for x
   x = log₃ 2 - 1
   
   The y intercept is given by
   (0 , f(0)) = (0,3^{(0 + 1)} - 2) = (0 , 1).
   
   
   
4. So far we have the domain, range, x and y intercepts and the horizontal asymptote. We need extra points.
   (-2 , f(-2)) = (-2, 3^{(-2 + 1)} - 2) = (4 , 1/3-2) = (4 , -1.67)
   
   
   (-4 , f(-4)) = (-4, 3^{(-4 + 1)} - 2) = (-4 , 2^-3) = (-4 , -1.99)
   
   Let us now use all the above information to graph f.
   
   

   

   ---