Session 04-02 - Power, Root & Rational Functions

Section 04: Advanced Functions

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Review from Session 04-01

Work individually for 5 minutes, then we discuss

  1. Determine the end behavior of \(P(x) = -3x^4 + 2x^2 - 7\)

  2. Given \(Q(x) = 2(x - 1)^3(x + 2)\), identify all zeros and their multiplicities and describe what they mean

  3. If a polynomial has degree 5, what is the maximal number of turning points it can have?

  4. Sketch \((x - 2)(x + 1)^2\) on a number line

Homework Discussion - 15 Minutes

Your questions from Tasks 04-01

Focus on polynomial applications and factoring

  • Challenges with the Rational Root Theorem
  • Sketching polynomials from factored form
  • Interpreting multiplicity in graphs
  • Business context questions

. . .

Power functions will help us understand the individual components of polynomials!

Learning Objectives

Today’s Goals

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

  • Understand power functions with integer, fractional, and negative exponents
  • Determine domains of root functions and fractional powers
  • Sketch power function graphs without a calculator
  • Compare growth rates of different power functions
  • Apply power functions to economic models
  • Analyze rational functions (asymptotes, holes) and average-cost models

Power Functions with Integer Exponents

Definition and Basic Forms

The building blocks of polynomials

A power function has the form \[f(x) = ax^n \quad (a \neq 0,\; n \text{ real})\]

. . .

  • Integer powers: \(x^2, x^3, x^{-1}, x^{-2}\)
  • Root functions: \(\sqrt{x}, \sqrt[3]{x}\)
  • Fractional powers: \(x^{2/3}, x^{3/2}\)

. . .

Simpler than polynomials, but they reveal behaviors that explain all functions.

Comparing even and odd powers

Negative Integer Powers (Reciprocal)

For \(n > 0\): \(f(x) = x^{-n} = \frac{1}{x^n}\), not defined at \(x = 0\)

Think-Pair-Share

2 minutes individual, 3 minutes pairs, 2 minutes class discussion

Compare and contrast:

  1. \(f(x) = x^2\) and \(g(x) = x^{-2}\)
  2. \(f(x) = x^3\) and \(g(x) = x^{-3}\)

Consider: domain and range · symmetry · behavior near \(x = 0\) · behavior as \(x \to \pm\infty\)

Think-Pair-Share: Discussion

Root Functions

Functions with fractional exponents

Domain Restrictions

Critical concept for root functions

like \(\sqrt{x}\), \(\sqrt[4]{x}\):

  • Domain: \(x \geq 0\) only
  • Why? Even roots of negative numbers aren’t real
  • \(\sqrt{x^2} = |x|\) (always non-negative)
  • Range: \(y \geq 0\)

like \(\sqrt[3]{x}\), \(\sqrt[5]{x}\):

  • Domain: all real numbers
  • Can take cube root of negative numbers
  • Range: all real numbers

Fractional Powers

General Fractional Exponents

Combining powers and roots

For \(f(x) = x^{m/n}\) where \(m, n\) are integers, \(n > 0\):

\[x^{m/n} = (x^{1/n})^m = \sqrt[n]{x^m}\]

. . .

Domain depends on \(n\):

  • If \(n\) is odd: all real numbers (usually)
  • If \(n\) is even: \(x \geq 0\) required

. . .

Example: \(f(x) = x^{3/2} = \sqrt{x^3} = (\sqrt{x})^3\)

  • Domain: \(x \geq 0\) (because of square root), grows faster than \(\sqrt{x}\) but slower than \(x^2\)

Comparing Growth Rates

Break - 10 Minutes

Economic Applications

Economies of Scale

Cost functions with fractional powers

Many production processes exhibit economies of scale:

\[C(x) = 500 + 50x^{0.7}\]

where \(x\) is production quantity (thousands)

. . .

Question: Any idea why \(x^{0.7}\)?

. . .

Exponent < 1 means cost grows slower than production!

Economies of Scale II

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  C_average = C_total / x  # Now both arrays have same length

Allometric Growth I

Biological and economic scaling

Many relationships follow power laws:

  • Biology: Brain mass ∝ (Body mass)\(^{0.75}\)
  • Economics:
    • City infrastructure costs ∝ (Population)\(^{0.85}\)
    • Company revenue ∝ (Number of employees)\(^{1.15}\)

. . .

Have you ever seen ∝ before? That’s the proportionality symbol (∝), which means “is proportional to” in mathematics.

Allometric Growth II

Surface Area and Volume

Geometric Power Relationships

Why packaging costs don’t scale linearly

Example: Doubling box dimensions

  • Surface area increases by factor of \(2^2 = 4\)
  • Volume increases by factor of \(2^3 = 8\)
  • Material cost (surface) vs. capacity (volume)

. . .

This explains why larger packages have lower cost per unit volume. The discount has a mathematical basis!

Graphing Techniques

Sketching Power Functions

A systematic approach

Steps to sketch \(f(x) = ax^n\) or \(f(x) = ax^{m/n}\):

Determine domain

  • Negative exponents: exclude \(x = 0\)
  • Even roots: require \(x \geq 0\)

Find key points

  • Always passes through \((1, a)\) if in domain
  • Check \((0, 0)\) if applicable

Analyze end behavior

  • Positive exponents: consider even/odd
  • Negative exponents: approach axes

Check symmetry

  • Even exponents: y-axis symmetry
  • Odd exponents: origin symmetry

Practice: Sketch Without Calculator

Work together

Sketch: \(f(x) = 2x^{2/3}\)

Sketch: The Result

Practice Domain and Range Analysis

Work together and discuss

For each function, determine the domain and range, then sketch a rough graph:

  1. \(f(x) = 3x^{1/4}\)

  2. \(g(x) = -2x^{-1}\)

  3. \(h(x) = x^{3/5}\)

  4. \(p(x) = 4 - x^{1/2}\)

Coffee Break - 15 Minutes

Combining Power Functions

Sums and Products

Building complex models

Real-world phenomena often combine power functions:

Total Cost with Multiple Effects:

\[C(x) = 1000x^{0.5} + 50x + 0.1x^2\]

  • \(x^{0.5}\): Setup costs (economies of scale)
  • \(x\): Linear variable costs
  • \(x^2\): Capacity constraints (diseconomies)

Rational Functions

Structure of Rational Functions

A rational function has the form:

\[f(x) = \frac{P(x)}{Q(x)}\]

where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \neq 0\)

  1. Domain: All real numbers except where \(Q(x) = 0\)
  2. Zeros: Where \(P(x) = 0\) (and \(Q(x) \neq 0\))
  3. Vertical Asymptotes: Where \(Q(x) = 0\) (canceling common factors)
  4. Holes: Where both \(P(x) = 0\) and \(Q(x) = 0\) (canceled factors)
  5. Horizontal/Oblique Asymptotes: Determined by degree comparison

What Are Asymptotes?

An asymptote is a line a function approaches

  • Think of it like a boundary the graph gets infinitely close to
  • Vertical asymptotes: Never crossed or touched (undefined there)
  • Horizontal/oblique asymptotes: Can be crossed at finite x-values, but approached as \(x \to \pm\infty\)
  • Three types: vertical, horizontal, and oblique (slanted)

Vertical Asymptotes

Occur where the denominator equals zero (and numerator doesn’t)

Mathematical definition: \[\lim_{x \to a^-} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = \pm\infty\]

  • The function “blows up” (goes to \(\infty\) or \(-\infty\))
  • Graph has a vertical line at \(x = a\)
  • Function is undefined at this point
  • Example: \(f(x) = \frac{1}{x}\) has vertical asymptote at \(x = 0\)

Horizontal Asymptotes

Three cases, based on the degrees of \(P(x)\) and \(Q(x)\)

\[f(x) = \frac{2x + 1}{x^3 - 5}\]

  • Denominator grows faster
  • Horizontal asymptote: \(y = 0\)
  • The function approaches zero

\[f(x) = \frac{3x^2 + 2x - 1}{2x^2 + 5}\]

  • Both grow at same rate
  • Horizontal asymptote: \(y = \frac{3}{2}\) (ratio of leading coefficients)

\[f(x) = \frac{x^3 + 2x}{x^2 - 1}\]

  • Numerator grows faster
  • No horizontal asymptote
  • May have an oblique (slanted) asymptote instead

Oblique (Slanted) Asymptotes

When degree of P exceeds degree of Q by exactly 1

  1. Perform polynomial long division1: \(\frac{P(x)}{Q(x)} = L(x) + \frac{R(x)}{Q(x)}\)
  2. The quotient \(L(x)\) (a linear function) is the oblique asymptote
  3. As \(x \to \pm\infty\), the remainder term \(\frac{R(x)}{Q(x)} \to 0\)

. . .

Example: \(f(x) = \frac{x^2 + 1}{x - 1} = \frac{x^2 + 1}{x - 1} = x + 1 + \frac{2}{x-1}\)

. . .

Oblique asymptote: \(y = x + 1\)

Holes vs. Asymptotes

Critical distinction when factors cancel!

  • Factor appears in both numerator and denominator
  • Example: \(f(x) = \frac{(x-2)(x+1)}{(x-2)(x+3)}\)
  • Factor \((x-2)\) cancels
  • Hole at \(x = 2\), not an asymptote!
  • Simplified: \(f(x) = \frac{x+1}{x+3}\), \(x \neq 2\)
  • Factor appears only in denominator
  • \(f(x) = \frac{x+1}{x+3}\)
  • Factor \((x+3)\) doesn’t cancel
  • Vertical asymptote at \(x = -3\)
  • Function undefined, goes to \(\pm\infty\)

. . .

Always factor completely and cancel common factors before identifying asymptotes!

Examples for Rational Functions

Asymptote Rules

Systematic Approach for Finding Asymptotes

Step 1: Factor completely \[f(x) = \frac{P(x)}{Q(x)} = \frac{\text{factored form}}{\text{factored form}}\]

Step 2: Cancel common factors → These create holes

Step 3: Vertical asymptotes → Remaining factors in denominator

Step 4: Horizontal/Oblique asymptotes → Compare degrees

Asymptote Analysis Challenge

3 minutes individual, 2 minutes pair discussion, 2 minutes class share

Analyze the function: \(f(x) = \frac{x^2 - x - 6}{x^2 - 4}\)

Your tasks:

  1. Factor numerator and denominator
  2. Identify any holes
  3. Find all asymptotes
  4. Determine x and y intercepts
  5. Sketch a rough graph

Asymptote Analysis

Business Application - Average Cost

Average Cost Functions

In business, the average cost per unit is:

\[AC(x) = \frac{\text{Total Cost}}{\text{Quantity}} = \frac{C(x)}{x} = \frac{F + vx}{x} = \frac{F}{x} + v\]

  • \(F\) = Fixed costs
  • \(v\) = Variable cost per unit
  • \(x\) = Number of units

. . .

Do you get the idea here?

Key Properties

These functions often have the same properties:

  • Vertical asymptote at \(x = 0\)
  • Horizontal asymptote at \(y = v\)
  • Always decreasing for \(x > 0\) (economies of scale)
  • Minimum average cost approaches \(v\) as \(x \to \infty\)

. . .

Let’s see an example!

Visualization of Average Cost

Manufacturing Analysis

Work through this business scenario and then we compare

A company has fixed costs of $5000 per month and variable costs of $20 per unit.

  1. Write the average cost function
  2. Find the horizontal asymptote and interpret it
  3. How many units minimize average cost to within $5 of the minimum?
  4. Graph the function

Guided Practice - 25 Minutes

Individual Exercise Block I

Work alone for 5 minutes, then discuss for 5 minutes

Problem 1: A company’s profit function combines multiple effects: \[P(x) = -2x^3 + 15x^2 + 100\sqrt{x} - 500\]

where \(x\) is production level (hundreds of units), \(x > 0\)

  1. Identify each term’s economic interpretation
  2. Calculate \(P(4)\) and \(P(9)\)
  3. Which term dominates for large \(x\)? What does this tell management?

Individual Exercise Block II

Work alone for 5 minutes, then discuss for 5 minutes

Problem 2: Compare growth rates for large values:

  1. Which grows faster: \(f(x) = x^{1/2}\) or \(g(x) = x^{1/3}\)?
  2. Which grows faster: \(f(x) = x^{3/2}\) or \(g(x) = x^2\)?
  3. Order from slowest to fastest growth: \(x^{1/2}, x, x^{3/2}, x^2\)

Individual Exercise Block III

Work alone for 5 minutes, then discuss for 5 minutes

Problem 3: A technology company’s average cost per unit is: \[AC(x) = \frac{50000}{x} + 100 + 0.01x\]

where \(x\) is units produced.

  1. Identify the power function in each term
  2. What happens to average cost as \(x \to 0^+\)?
  3. What happens to average cost as \(x \to \infty\)?
  4. Graph the behavior conceptually

Spot the Error

Can you find what’s wrong? Work with your neighbor

Time allocation: 5 minutes to find errors, 5 minutes to discuss

Student work:

  1. “\(\sqrt{x^2} = x\) for all \(x\)”

  2. “The function \(f(x) = x^{-1/2}\) has domain \(x > 0\)”

  3. “Since \(x^{2/3} = \sqrt[3]{x^2}\), the domain is \(x \geq 0\)”

  4. “\(x^{1.5}\) grows faster than \(x^2\) because 1.5 is complicated”

. . .

Not everything has to be wrong!

Wrap-Up & Key Takeaways

Key Takeaways

Today’s essential concepts

  • Power functions are the building blocks of polynomials
  • Domain restrictions come from mathematical necessity
  • Growth rate comparisons guide long-term planning
  • Fractional powers create realistic economic models

Final Assessment

5 minutes - Individual work

A manufacturing company’s cost per unit follows: \[C(x) = 10000x^{-0.5} + 50 + 2x^{0.5}\]

where \(x\) is the number of units produced (in thousands).

  1. What is the domain of this function in the business context?

  2. Identify each term’s economic meaning

  3. What happens to cost per unit as production increases dramatically?

  4. Which term represents economies of scale?

Next Session Preview

Session 04-03: Exponential & Logarithmic Functions

Moving from power to exponential growth

  • Exponential growth and decay, doubling time and half-life
  • The natural exponential \(e^x\), compound interest, continuous growth
  • Exponential vs. power function growth (critical comparison!)
  • Logarithmic functions, semi-log and log-log scales

. . .

Complete Tasks 04-02!

Footnotes

  1. No worries, no need to learn long division. This is just for the sake of completeness.↩︎