1. Introduction
Because we have already studied linear and quadratic functions, we can now begin working with exponential functions. In this lesson, we will explore the domain and range of exponential functions, as well as practice evaluating exponential functions. In particular, we will practice identifying the domain and range of functions that represent a real-world situation.
Domain and Range of Exponential Functions

2. Introduction, continued
For example, say you deposit $2,000 into a savings account that pays 3% interest every year, and you withdraw all of your money after 5 full years. The amount of money in your account grows according to the function f(x) = 2000 • 1.03x, where x is measured in years. What are the possible values of x and f(x) for this situation?
Domain and Range of Exponential Functions

3. Introduction, continued
You wouldn’t use negative values for x, since the situation starts when you deposit the money, at x = 0. If you took all of your money out of the account after 5 years, you also wouldn’t use values of x greater than 5, because the situation no longer applies after this time. Also, you would only look at whole numbers for x, since the value of the account increases only once per year. You would also only consider f(x) values for whole values of x, and you wouldn’t look at f(x) values for x < 0 or x > 5. In other words, the domain of the function is {0, 1, 2, 3, 4, 5}, and the range of the function is {f(0), f(1), f(2), f(3), f(4), f(5)}.
Domain and Range of Exponential Functions

4. Introduction, continued
Because of the limits of the real-world situation, this domain and range is significantly different from the mathematical domain and range of f(x) = 2000 • 1.03x. Without the real-world context, the domain of f(x) = 2000 • 1.03x is all real numbers, and the range is f(x) > 0.
Domain and Range of Exponential Functions

5. Key Concepts
Recall that the domain of a function is the set of all inputs, or x-values, that are valid for the function.
Recall that the range of a function is the set of all outputs, or f(x) values, that are valid for the function.
Recall that to evaluate a function f(x), you must substitute values of x for each variable x in the function.
Domain and Range of Exponential Functions

6. Key Concepts, continued
An exponential function is a function that has a variable in the exponent. The general form is f(x) = abx + k, where a is the initial value, b ≠ 1 is the base, and k is a constant value.
A horizontal asymptote is a horizontal line that a function gets closer and closer to as x increases or decreases without bound. Every exponential function with a domain of all real numbers has a horizontal asymptote. The asymptote will be the line y = k, where k is the constant in the exponential function.
Domain and Range of Exponential Functions

7. Key Concepts, continued
The domain of an exponential function with no restrictions is all real numbers.
The range of an exponential function with no restrictions is a subset of the real numbers. One end of the range will be k; the other end of the range will be infinite.
For example, find the range of the function
f(x) = –1 • 3x + 1.
Domain and Range of Exponential Functions

8. Key Concepts, continued
As x increases, the function becomes more and more negative. You can determine this by evaluating the function for increasingly large values of x: x
f(x) 1 –2 5
–242 10
–59,048
Domain and Range of Exponential Functions

9. Key Concepts, continued
As x decreases, the function gets closer and closer to 1, although it will never reach 1 exactly: x
f(x) –1 – –5
–10
Domain and Range of Exponential Functions

10. Key Concepts, continued
The range of the function is (–∞, 1), or f(x) < 1.
You can also identify the domain and range of an exponential function by looking at its graph.
The domain will be all valid x-values on the graph. If the function continues off the edge of the graph (if the graph has arrows on the ends), the domain is all real numbers.
The range will be all valid y-values on the graph. If the function continues off the edge of the graph, the range will be all f(x) values spanned between the asymptote value and infinity. Note that the infinity could be negative, depending on the shape of the graph.
Domain and Range of Exponential Functions

11. Common Errors/Misconceptions
thinking that f(x) means f • x
not understanding how to use the domain to find the range
confusing the domain and range of a function
Domain and Range of Exponential Functions

12. Guided Practice Example 1
Use the following graph to identify the domain and range of the function f(x) = 2x.
Domain and Range of Exponential Functions

13. Guided Practice: Example 1, continued Identify the domain.
The domain is the set of x-values that are valid for the function. This graph goes on infinitely; so, the domain can be any real x-value.
Domain: {all real numbers}
Domain and Range of Exponential Functions

14. ✔ Guided Practice: Example 1, continued Identify the range.
The range is the set of y-values that are valid for the function.
The range will never go below 0. In fact, the range never actually reaches 0. The upper end of the range, however, is limitless.
Range: {y > 0} ✔
Domain and Range of Exponential Functions

15. Guided Practice: Example 1, continued 15
Domain and Range of Exponential Functions

16. Guided Practice Example 3
A population of bacteria doubles every 4 hours according to the function f(x) = 32 • 20.25x. Find the domain and range of the function.
Domain and Range of Exponential Functions

17. Guided Practice: Example 3, continued Identify the domain.
An exponential function has a domain of all real numbers, but we must consider the context of the problem. Because time cannot be negative, the domain is all real numbers greater than or equal to 0.
So, the domain of f(x) = 32 • 20.25x is [0, ∞), or x ≥ 0.
Domain and Range of Exponential Functions

18. Guided Practice: Example 3, continued Identify the range.
The range is the set of f(x) values that are valid for the function. Because the function gets bigger as x increases, the smallest value will occur at x = 0. The upper end of the range is limitless.
Domain and Range of Exponential Functions

19. Guided Practice: Example 3, continued
To find the lower end of the range, evaluate f(x) = 32 • 20.25x for x = 0:
f(x) = 32 • 20.25x
Original function
f(0) = 32 • 20.25(0)
Substitute 0 for x.
f(0) = 32 • 20
Simplify.
f(0) = 32 • 1
Evaluate the exponent.
f(0) = 32
Domain and Range of Exponential Functions

20. ✔ Guided Practice: Example 3, continued
Therefore, the range of f(x) = 32 • 20.25x is (32, ∞), or f(x) ≥ 32. ✔
Domain and Range of Exponential Functions

21. Guided Practice: Example 3, continued
Domain and Range of Exponential Functions