# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/exponential-logarithm/function-definition Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/exponential-logarithm/function-definition/en.mdx Learn exponential function definition f(x)=a^x with domain, range & properties. Understand base conditions and real-world applications. --- ## Exponential Function An exponential function is a function expressed in the form: ```math f(x) = n \times a^x ``` with the conditions: - $$a$$ is the base number, where $$a > 0$$ and $$a \neq 1$$ - $$n$$ is a non-zero real number - $$x$$ is any real number Visible text: - is the base number, where and - is a non-zero real number - is any real number Exponential functions have a special characteristic where the variable $$x$$ is in the exponent position. This is what distinguishes exponential functions from ordinary algebraic functions. In exponential functions, small changes in the value of $$x$$ can result in very large changes in the function's output. Visible text: Exponential functions have a special characteristic where the variable is in the exponent position. This is what distinguishes exponential functions from ordinary algebraic functions. In exponential functions, small changes in the value of can result in very large changes in the function's output. ## Properties of Exponential Functions The exponential function $$f(x) = a^x$$ (for $$n = 1$$) has several important properties: Visible text: The exponential function (for ) has several important properties: 1. The domain of the function is all real numbers ($$\mathbb{R}$$) 2. The range of the function is all positive numbers ($$\mathbb{R}^+$$) 3. It intersects the vertical axis at point $$(0, 1)$$ because $$a^0 = 1$$ 4. The function is always positive for all values of $$x$$ because $$a > 0$$ 5. If $$a > 1$$, the function increases (monotonically increasing) 6. If $$0 < a < 1$$, the function decreases (monotonically decreasing) Visible text: 1. The domain of the function is all real numbers () 2. The range of the function is all positive numbers () 3. It intersects the vertical axis at point because 4. The function is always positive for all values of because 5. If , the function increases (monotonically increasing) 6. If , the function decreases (monotonically decreasing) ## Special Cases of Exponential Functions ### When the Base Is One If $$a = 1$$, then: Visible text: If , then: ```math f(x) = n \times 1^x = n ``` The value of $$1^x$$ is always $$1$$ for any value of $$x$$. As a result, the function becomes a constant function $$f(x) = n$$, no longer an exponential function. Its graph will be a horizontal line intersecting the vertical axis at point $$(0, n)$$. Visible text: The value of is always for any value of . As a result, the function becomes a constant function , no longer an exponential function. Its graph will be a horizontal line intersecting the vertical axis at point . Component: FunctionChart Props: - a: 1 - p: 1 - title: Constant Function - description: Line is always horizontal constant at $$y = 1$$. Visible text: Line is always horizontal constant at . ### When the Base Is Zero If $$a = 0$$, then: Visible text: If , then: ```math f(x) = n \times 0^x ``` - For $$x > 0$$, the value of $$0^x = 0$$ so $$f(x) = 0$$ - For $$x = 0$$, the value of $$0^0$$ is undefined - For $$x < 0$$, the value of $$0^x$$ is undefined Visible text: - For , the value of so - For , the value of is undefined - For , the value of is undefined This function is no longer an exponential function but rather constant at $$f(x) = 0$$ for $$x > 0$$. Then, because $$0^0$$ and $$0^x$$ for $$x < 0$$ are undefined, this function does not meet the definition of an exponential function. Visible text: This function is no longer an exponential function but rather constant at for . Then, because and for are undefined, this function does not meet the definition of an exponential function. Component: FunctionChart Props: - a: 0 - p: 1 - title: Constant Function - description: Line is always horizontal constant at $$y = 0$$, but undefined for $$x = 0$$. Visible text: Line is always horizontal constant at , but undefined for . ## Examples of Exponential Functions Here are some examples of exponential functions: 1. $$f(x) = 4^x$$ This function has base number $$a = 4$$ and $$n = 1$$. Since $$a > 1$$, this function is monotonically increasing. The function value will get larger as $$x$$ increases. For example, $$f(0) = 4^0 = 1$$, $$f(1) = 4^1 = 4$$, $$f(2) = 4^2 = 16$$. 2. $$f(x) = 3^{x+1}$$ This function can be rewritten as $$f(x) = 3 \times 3^x$$ with base number $$a = 3$$ and $$n = 3$$. The graph of this function is also monotonically increasing, and the function value will get larger as $$x$$ increases. For example, $$f(0) = 3^{0+1} = 3^1 = 3$$, $$f(1) = 3^{1+1} = 3^2 = 9$$. 3. $$f(x) = 5^{2x-1}$$ This function has base number $$a = 5$$ with exponent $$2x-1$$. The function value will change more rapidly because the coefficient of $$x$$ is $$2$$. For example, $$f(0) = 5^{2(0)-1} = 5^{-1} = \frac{1}{5}$$, $$f(1) = 5^{2(1)-1} = 5^1 = 5$$. 4. $$f(x) = 0.5^x$$ This function has base number $$a = 0.5$$ where $$0 < a < 1$$. This function is monotonically decreasing. The function value will get smaller as $$x$$ increases. For example, $$f(0) = 0.5^0 = 1$$, $$f(1) = 0.5^1 = 0.5$$, $$f(2) = 0.5^2 = 0.25$$. Visible text: 1. This function has base number and . Since , this function is monotonically increasing. The function value will get larger as increases. For example, , , . 2. This function can be rewritten as with base number and . The graph of this function is also monotonically increasing, and the function value will get larger as increases. For example, , . 3. This function has base number with exponent . The function value will change more rapidly because the coefficient of is . For example, , . 4. This function has base number where . This function is monotonically decreasing. The function value will get smaller as increases. For example, , , . ## Applications of Exponential Functions Exponential functions are widely used in everyday life and various fields: 1. **Population Growth**: The number of bacteria reproducing can be modeled with an exponential function $$P(t) = P_0 \times 2^{t/n}$$ where $$P_0$$ is the initial number, $$t$$ is time, and $$n$$ is the time required for the population to double. 2. **Compound Interest**: If someone saves money with compound interest, the amount of savings after $$t \text{ years}$$ can be calculated with $$A(t) = P \times (1 + r)^t$$ where $$P$$ is the initial principal and $$r$$ is the interest rate. 3. **Radioactive Decay**: The amount of radioactive substance remaining after $$t \text{ years}$$ can be calculated with $$A(t) = A_0 \times 0.5^{t/h}$$ where $$A_0$$ is the initial amount and $$h$$ is the half-life. 4. **Virus Spread**: The spread of disease in a population often follows an exponential model in the early phase. Visible text: 1. **Population Growth**: The number of bacteria reproducing can be modeled with an exponential function where is the initial number, is time, and is the time required for the population to double. 2. **Compound Interest**: If someone saves money with compound interest, the amount of savings after can be calculated with where is the initial principal and is the interest rate. 3. **Radioactive Decay**: The amount of radioactive substance remaining after can be calculated with where is the initial amount and is the half-life. 4. **Virus Spread**: The spread of disease in a population often follows an exponential model in the early phase.