Domain: {all real numbers} Range: {f(x) | f(x)>-1}
Practice makes perfect
Let's begin by graphing the function. Then we will state its domain and range.
Graphing the Function
We want to draw a graph of the given exponential function.
f(x)=3 ( 1/4 )^(x+3)-1
Because the base of our function is less than 1, we know that it is an exponential decay function. Let's start by making a table of values.
x
3(1/4)^(x+3)-1
y=3(1/4)^(x+3)-1
- 4
3(1/4 )^(- 4 +3)-1
11
- 3
3(1/4 )^(- 3 +3)-1
2
- 2
3(1/4 )^(- 2+3)-1
- 0.25
- 1
3(1/4 )^(- 1+3)-1
- 0.8125
0
3(1/4 )^(0+3)-1
- 0.953 ...
1
3(1/4 )^(1+3)-1
-0.988 ...
All of the ordered pairs ( - 4, 11), ( - 3, 2), ( - 2, - 0.25), ( - 1, - 0.8125), ( 0, - 0.953), and ( 1, -0.988), belong to the graph of our function. Now, we will plot and connect these points with a smooth curve.
Determining the Domain and Range
Unless a restriction is specifically stated, the domain of any exponential function is all real numbers. The graph of our function is above the line y=-1, so the range is all real numbers that are greater than -1.
Domain:& { all real numbers }
Range:& {f(x) | f(x) >-1 }
