An exponential function is a mathematical formula that describes a quantity that changes rapidly over time. Unlike a linear function, which increases or decreases at a steady rate, an exponential function's rate of change speeds up as the value grows. This characteristic makes exponential functions ideal for modelling real-world situations such as compound interest, population growth, radioactive decay, and viral spread.
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An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It is written as:
f(x) = a × bx
Where:
a is a constant and a ≠ 0
b is the base of the exponential (b > 0 and b ≠ 1)
x is the exponent (independent variable)
Exponential functions are unique because the variable appears in the exponent, while in polynomial functions, the variable is the base.
Growth: If b > 1, the function indicates exponential growth.
Decay: If 0 < b < 1, the function shows exponential decay.
Understanding what an exponential function is helps solve real-world problems where quantities double or halve over time.
The general formula for exponential functions is:
f(x) = a × b x
Where:
f(x) is the output
a is the initial value
b is the base (the growth or decay rate)
x is the exponent
Common Forms:
Type Formula Condition
Exponential Growth f(x) = a × (1 + r) x r > 0
Exponential Decay f(x) = a × (1 - r) x 0 < r < 1
Natural Exponential f(x) = a × e x e ≈ 2.718
Understanding the exponential function formula is vital for using these functions in various mathematical models.
Graphs of exponential functions have distinct shapes based on whether they show growth or decay.
Key Features of the Graph:
Passes through the point (0, a)
Increases rapidly for exponential growth
Decreases toward zero for exponential decay
The graph gets very close to the x-axis but never actually touches it.
Domain: all real numbers
Range: depends on the sign of a (see the next section)
Example:
For f(x) = 2 x
The graph increases rapidly as x increases.
It passes through (0, 1), (1, 2), (2, 4), etc.
Example:
For f(x) = (1/2) x
The graph decays and gets closer to zero as x increases.
These graphs of exponential functions help visualise how quickly quantities grow or decay in real situations.
Rapid increase or decrease
Always positive (if a > 0)
The graph keeps getting closer to the x-axis but never reaches it.”
Can be reflected across the x-axis if a < 0
The exponent must be a variable (e.g., x), not a constant
Mastering these characteristics improves your understanding of exponential functions in different contexts.
Let’s look at some real-world examples of exponential functions that show growth and decay.
Formula: A = P × (1 + r)t
Step 1: P = 1000, r = 5% = 0.05, t = 3
Step 2: A = 1000 × (1 + 0.05)3
Step 3: A = 1000 × (1.05)3
Step 4: 1.05 × 1.05 × 1.05 = 1.157625
Step 5: A = 1000 × 1.157625 ≈ 1157.63
Formula: P(t) = 500 × 2t
Step 1: Initial population = 500, t = 3 hours
Step 2: P(3) = 500 × 23
Step 3: 23 = 2 × 2 × 2 = 8
Step 4: P(3) = 500 × 8 = 4000
Formula: N(t) = 1000 × (1/2)t
Step 1: Initial quantity = 1000, t = 2 half-lives
Step 2: N(2) = 1000 × (1/2)2
Step 3: (1/2)2 = 1/2 × 1/2 = 1/4
Step 4: N(2) = 1000 × 1/4 = 250
Formula: V(t) = 10 × 2t
Step 1: Initial infected = 10, t = 5 days
Step 2: V(5) = 10 × 25
Step 3: 25 = 2 × 2 × 2 × 2 × 2 = 32
Step 4: V(5) = 10 × 32 = 320
Formula: f(x) = 3 × e x
Step 1: Base constant = 3, x = 1, e ≈ 2.718
Step 2: f(1) = 3 × e1 = 3 × 2.718
Step 3: f(1) ≈ 8.154
Understanding the domain and range of exponential functions is essential for graphing and solving problems.
Domain:
The domain of all exponential functions is (-∞, ∞).
This means x can take any real value.
Range:
Depends on the sign of a.
If a > 0, the range is (0, ∞).
If a < 0, the range is (-∞, 0).
Function
Domain Range
f(x) = 2 x (-∞, ∞) (0, ∞)
f(x) = -3 x (-∞, ∞) (-∞, 0)
Special Notes:
The graph never touches the x-axis.
The y-values get closer to but never reach the horizontal asymptote (usually y = 0).
Understanding domain and range tells us how far the graph stretches across (x-values) and how high or low it goes (y-values).
Some people confuse exponentials with polynomials, but exponentials grow much faster for large numbers.
It never really reaches zero; it approaches asymptotically.
The sign of a and the base value greatly affect the shape.
Exponentials grow faster than polynomials for large x.
They also model decay, which is common in nature and science.
Approximately 2.718 is the base of natural exponential functions.
The famous mathematician who introduced the constant e.
Many machine learning and computer algorithms use exponentials.
Population growth, decay, and even tree branching can be modelled exponentially.
Exponents help us write really big or tiny numbers easily in science, like the size of planets or atoms
Question: Evaluate f(x) = 2 x when x = 4
Step 1: Substitute x = 4 → f(4) = 24
Step 2: Expand → 2 × 2 × 2 × 2
Step 3: Multiply → 16
Final Answer: f(4) = 16
Question: Is f(x) = 3 x exponential growth or decay?
Step 1: Base = 3
Step 2: Since 3 > 1 → it is growth
Step 3: If there is a negative sign in front, the graph flips below x-axis
Final Answer: Exponential Growth
Question: What is the domain and range of f(x) = –2 x ?
Step 1: Domain of exponential function is all real numbers → (–∞, ∞)
Step 2: Since the function is negative, outputs are below 0
Step 3: Range = (–∞, 0)
Final Answer: Domain = (–∞, ∞), Range = (–∞, 0)
Question: f(x) = 500×(1/2) x . Find f(3).
Step 1: Substitute x = 3 → f(3) = 500× (1/2)3
Step 2: (1/2)3 = 1/8
Step 3: Multiply → 500 × (1/8) = 62.5
Final Answer: f(3) = 62.5
Question: Find the output of f(x) = e x when x = 1
Step 1: Substitute x = 1 → f(1) = e1
Step 2: Simplify → f(1) = e
Step 3: Approximate → e ≈ 2.718
Final Answer: f(1) ≈ 2.718
In conclusion, exponential functions play a crucial role in mathematics and various real-world applications. They help model population growth, radioactive decay, financial changes, and technology trends. By mastering the exponential function formula, examining graphs of exponential functions, applying real-world examples, and understanding their domain and range, students can build a strong foundation in this powerful mathematical concept. Knowing what an exponential function is and how it operates opens up opportunities for deeper learning in algebra, calculus, and data science. With proper understanding and practice, exponential functions become more than just a textbook topic; they become practical tools for solving everyday problems.
Answer: An exponential function is a mathematical function of the form f(x) = a×r x , where a is not equal to 0 and r is greater than 0.
Answer: The key rules are that the base r must be positive, r cannot equal 1, and the exponent is a variable.
Answer: No, 2x is a linear function. An exponential function has the variable in the exponent, like 2 x .
Answer: In an exponential function f(x) = a×r x , r is the constant base or growth or decay factor.
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