Exponential Functions

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# Exponential Functions ## 🎯 Learning Objectives By the end of this topic, you will be able to: 1. **Define** exponential functions and distinguish them from polynomials 2.

Exponential Functions

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define exponential functions and distinguish them from polynomials
Apply the laws of exponents
Graph exponential functions for $a > 1$ and $0 < a < 1$
Identify exponential growth vs. decay
Use the natural exponential function $e^x$
Apply the vertical and horizontal line tests to graphs

📋 Prerequisites

Functions — domain, range, function notation
Exponent rules — $a^m \cdot a^n = a^{m+n}$ , $(a^m)^n = a^{mn}$ , $a^0 = 1$
Polynomials — basic graphing concepts

📖 Core Content

6.1 Exponential vs. Polynomial

6.1.1 Intuition

A polynomial like

x^2

grows by multiplying the base (

x

) by itself. An exponential like

2^x

grows by multiplying the exponent (

x

) by 2 each step. The difference is dramatic:

x^2

gives

1, 4, 9, 16, 25, \ldots

while

2^x

gives

2, 4, 8, 16, 32, \ldots

. Eventually, exponentials outpace any polynomial.

💡 Why this matters: Exponentials model growth processes: population, compound interest, viral spread, machine learning training time (sometimes). Decay processes: radioactive decay, cooling, forgetting curves.

6.1.2 Definition

An exponential function has the form:

f(x) = a^x

where

a > 0

and

a \neq 1

is the base, and

x

is the exponent.

Base	Behavior	Example
$a > 1$	Exponential growth	$2^x$ , $3^x$ , $e^x$
$0 < a < 1$	Exponential decay	$(1/2)^x$ , $0.1^x$

Key property:

a^0 = 1

for any

a \neq 0

, so every exponential passes through

(0, 1)

6.2 Laws of Exponents

Law	Formula	Example
Product	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^4 = 2^7$
Quotient	$a^m / a^n = a^{m-n}$	$3^5 / 3^2 = 3^3$
Power	$(a^m)^n = a^{mn}$	$(5^2)^3 = 5^6$
Negative	$a^{-n} = 1/a^n$	$2^{-3} = 1/8$
Zero	$a^0 = 1$ (if $a \neq 0$ )	$7^0 = 1$
Fractional	$a^{m/n} = \sqrt[n]{a^m}$	$8^{2/3} = \sqrt[3]{8^2} = 4$
Distributive	$(ab)^n = a^n b^n$	$(2 \cdot 3)^2 = 36$
	$(a/b)^n = a^n / b^n$	$(4/2)^3 = 8$

6.3 Graphing Exponential Functions

6.3.1 Graph of $f(x) = a^x$ , $a > 1$ (Growth)

For

f(x) = 2^x

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$1/8$	$1/4$	$1/2$	$1$	$2$	$4$	$8$

Properties:

Domain: $\mathbb{R}$ (all real numbers)
Range: $(0, \infty)$ (always positive)
$y$ -intercept: $(0, 1)$
Horizontal asymptote: $y = 0$ (x-axis) as $x \to -\infty$
Increasing, convex (curving upward)

6.3.2 Graph of $f(x) = a^x$ , $0 < a < 1$ (Decay)

For

f(x) = (1/2)^x

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$8$	$4$	$2$	$1$	$1/2$	$1/4$	$1/8$

Properties:

Domain: $\mathbb{R}$ , Range: $(0, \infty)$
$y$ -intercept: $(0, 1)$
Horizontal asymptote: $y = 0$ as $x \to \infty$
Decreasing, convex (curving upward)

🔍 Note: $(1/2)^x = 2^{-x}$ . So decay functions are reflections of growth functions across the $y$ -axis. (Diagram)

6.4 The Natural Exponential Function

The natural exponential

f(x) = e^x

has base

e \approx 2.71828\ldots

, which is an irrational number (like

\pi

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828459045\ldots

Why $e$ ? It's the "natural" base for growth processes — continuous compounding, population growth, and it's its own derivative (key in calculus!). Example — Continuous Compounding: If you invest

100 at 5% annual interest compounded continuously, after

years:

A(t) = 100 \cdot e^{0.05t}$.

6.5 Vertical and Horizontal Line Tests

6.5.1 Vertical Line Test (VLT)

A graph represents a function iff no vertical line intersects it more than once. All exponential functions pass the VLT (they're functions).

6.5.2 Horizontal Line Test (HLT)

A function is injective (one-to-one) iff no horizontal line intersects its graph more than once. All exponential functions

f(x) = a^x

(with

a > 0

a \neq 1

) pass the HLT — they are injective. This means they have inverses (logarithms!).

6.6 Worked Examples

Example 6.1 (Growth): The population of a city grows at 3% per year. If the current population is 500,000, find the population after 10 years. Step 1:

P(t) = P_0 \cdot a^t

, where

a = 1 + 0.03 = 1.03

P_0 = 500000

Step 2:

P(10) = 500000 \cdot (1.03)^{10}

Step 3:

(1.03)^{10} \approx 1.3439

Step 4:

P(10) \approx 500000 \times 1.3439 = 671958

\boxed{\approx 672,000}

Example 6.2 (Decay): A radioactive isotope decays with half-life 5 years. If we start with 100g, how much remains after 20 years? Step 1: Half-life means

N(t) = N_0 \cdot (1/2)^{t/5}

Step 2:

N(20) = 100 \cdot (1/2)^{20/5} = 100 \cdot (1/2)^4 = 100 \cdot \frac{1}{16} = 6.25

\boxed{6.25 \text{ grams}}

Example 6.3 (Natural exponential): Simplify

e^{3x} \cdot e^{-2x} \cdot e^{0}

. Step 1:

e^{3x} \cdot e^{-2x} \cdot e^0 = e^{3x + (-2x) + 0} = e^{x}

\boxed{e^x}

📐 Key Formulas — Summary Table

Concept	Formula	Notes
Exponential function	$f(x) = a^x$ , $a > 0$ , $a \neq 1$
Growth	$a > 1$	Increasing
Decay	$0 < a < 1$	Decreasing
$e$	$e \approx 2.71828$	Natural base
Product	$a^m \cdot a^n = a^{m+n}$
Quotient	$a^m / a^n = a^{m-n}$
Power	$(a^m)^n = a^{mn}$
Negative	$a^{-n} = 1/a^n$
Fractional	$a^{m/n} = \sqrt[n]{a^m}$
HLT	Exponential passes → injective	Inverse exists (log)
Domain	$\mathbb{R}$
Range	$(0, \infty)$
$y$ -intercept	$(0, 1)$
Asymptote	$y = 0$	Horizontal

⚠️ Common Pitfalls

Pitfall 1: Confusing $x^n$ with $n^x$

x^n

is polynomial (base varies, exponent fixed);

n^x

is exponential (base fixed, exponent varies). Very different behavior!

Pitfall 2: Thinking $a^0 = 0$

Actually

a^0 = 1

(for

a \neq 0

). This is why every exponential passes through

(0, 1)

Pitfall 3: Forgetting the Domains Are All Reals

Exponential functions accept any real input, including irrationals.

2^{\sqrt{2}}

is well-defined.

Pitfall 4: Assuming $e$ is a Variable

e

is a constant (

\approx 2.718

), not a variable.

e^x

is the exponential function with base

e

📝 Practice Questions

Q1: Evaluate $3^{-2} \cdot 3^4$ .

Strategy Hint: Use product law.

$3^{-2+4} = 3^2 = 9$

$\boxed{9}$ Q2: Simplify $(2^3)^4 \div 2^7$ .

Strategy Hint: Power law then quotient law.

$(2^3)^4 = 2^{12}$ ; $2^{12} \div 2^7 = 2^5 = 32$

$\boxed{32}$ Q3: Graph $f(x) = 3^x$ and state its range.

Strategy Hint: Plot points for $x = -2, -1, 0, 1, 2$ .

Range: $(0, \infty)$

$\boxed{(0, \infty)}$ Q4: Is $f(x) = 2^x + 1$ injective?

Strategy Hint: Check HLT — horizontal lines intersect at most once?

Yes, $2^x$ is injective, adding 1 shifts it up but doesn't change injectivity.

$\boxed{\text{Yes}}$ Q5: Simplify $\frac{e^{2x} \cdot e^{-x}}{e^{3x}}$ .

Strategy Hint: Combine exponents in numerator, then subtract.

$= \frac{e^{x}}{e^{3x}} = e^{x-3x} = e^{-2x}$

$\boxed{e^{-2x}}$ Q6: A bacteria culture doubles every 3 hours. If initial count is 1000, find count after 12 hours.

Strategy Hint: $N(t) = N_0 \cdot 2^{t/3}$ .

$N(12) = 1000 \cdot 2^{4} = 1000 \cdot 16 = 16000$

$\boxed{16,000}$ Q7: Determine if $f(x) = \left(\frac{2}{3}\right)^x$ represents growth or decay.

Strategy Hint: Base = $2/3 \approx 0.667 < 1$ .

Decay.

$\boxed{\text{Decay}}$ Q8: Compute $(8)^{-2/3}$ .

Strategy Hint: $8^{-2/3} = \frac{1}{8^{2/3}} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{2^2} = \frac{1}{4}$ .

$\boxed{\frac{1}{4}}$ Q9: If $f(x) = 2^{x+1}$ , find $f(3)$ .

$f(3) = 2^{4} = 16$

$\boxed{16}$ Q10: The half-life of a drug in the body is 6 hours. If a patient takes 200mg, how much remains after 24 hours?

$N(24) = 200 \cdot (1/2)^{24/6} = 200 \cdot (1/2)^4 = 200 \cdot 1/16 = 12.5$

$\boxed{12.5 \text{ mg}}$ Q11: Write $\sqrt{2} \cdot \sqrt[3]{2}$ in the form $2^x$ .

$\sqrt{2} = 2^{1/2}$ , $\sqrt[3]{2} = 2^{1/3}$ . Product: $2^{1/2 + 1/3} = 2^{5/6}$

$\boxed{2^{5/6}}$ Q12: If a function fails the horizontal line test, does it pass the vertical line test?

Strategy Hint: These tests check different things.

HLT checks injectivity. VLT checks if it's a function. A function can fail HLT (not injective) but always passes VLT.

$\boxed{\text{Yes — it's still a function but not injective}}$

🔗 Cross-References

Next topic: Composite and Inverse Functions
Related: Logarithms — the inverse of exponentials
Across courses: BSMA1003 Maths 2 (exponential functions in differential equations); BSMA1002 Stats 1 (exponential distributions, growth models) Join Discord Previous4.0 Polynomial OperationsNext5.2 Composite & Inverse Functions

What this note is really saying

Exponential Functions

🎯 Learning Objectives

📋 Prerequisites

📖 Core Content

6.1 Exponential vs. Polynomial

6.1.1 Intuition

6.1.2 Definition

6.2 Laws of Exponents

6.3 Graphing Exponential Functions

6.3.1 Graph of f(x)=axf(x) = a^xf(x)=ax, a>1a > 1a>1 (Growth)

6.3.2 Graph of f(x)=axf(x) = a^xf(x)=ax, 0<a<10 < a < 10<a<1 (Decay)

6.4 The Natural Exponential Function

6.5 Vertical and Horizontal Line Tests

6.5.1 Vertical Line Test (VLT)

6.5.2 Horizontal Line Test (HLT)

6.6 Worked Examples

📐 Key Formulas — Summary Table

⚠️ Common Pitfalls

Pitfall 1: Confusing xnx^nxn with nxn^xnx

Pitfall 2: Thinking a0=0a^0 = 0a0=0

Pitfall 3: Forgetting the Domains Are All Reals

Pitfall 4: Assuming eee is a Variable

📝 Practice Questions

🔗 Cross-References

Intelligence Hub

6.3.1 Graph of $f(x) = a^x$ , $a > 1$ (Growth)

6.3.2 Graph of $f(x) = a^x$ , $0 < a < 1$ (Decay)

Pitfall 1: Confusing $x^n$ with $n^x$

Pitfall 2: Thinking $a^0 = 0$

Pitfall 4: Assuming $e$ is a Variable