Session 04-02 - Power, Root & Rational Functions

Section 04: Advanced Functions

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

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

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

Integer Powers

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

Two panels. Even powers x², x⁴ and x⁶ are U-shaped and symmetric about the y-axis. Odd powers x, x³ and x⁵ run from bottom left to top right through the origin. All pass through (1, 1); the higher the power, the flatter the curve near 0 and the steeper beyond 1.

Negative Integer Powers

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

Two panels of negative powers, each with a gap at x = 0, where the curves shoot off along the y-axis. Left: 1/x (solid) and 1/x³ (dashed) are negative for negative x and positive for positive x. Right: 1/x² (solid) and 1/x⁴ (dashed) are positive on both sides. All curves approach zero as x moves away from 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

Two panels of root functions. Left: the square, fourth and sixth roots start at the origin and exist only for x ≥ 0; they rise steeply at first and then flatten, the higher roots sooner. Right: the cube root and the fifth root are defined for all x and pass through the origin, with negative values for negative x.

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

Curves of x^(1/2), x, x^(3/2), x² and x³ for x from 0 to 3. All meet at (1, 1); beyond that point, the larger the exponent, the faster the curve rises. The square root is labelled as the slowest growth, x³ as the fastest.

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

Two panels for the cost function C(x) = 500 + 50x^0.7. Left: total cost rises with production, but ever more slowly. Right: the average cost per unit drops steeply at first and then levels off as production grows.

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

Power-law curves against size from 1 to 100, compared with a dashed straight line for the exponent 1. The curves with exponents 0.75 and 0.85 bend below the line and fall further behind as size grows; the curve with exponent 1.15 bends above it and pulls away.

Area & 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

Graph of f(x) = 2x^(2/3) for x from -3 to 3: two branches, symmetric about the y-axis, that meet in a sharp point (cusp) at the origin and rise ever more slowly outward. The points (-1, 2), (0, 0) and (1, 2) are marked.

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

Break

Combining Powers

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 division (as in 04-01): \(\frac{P(x)}{Q(x)} = L(x) + \frac{R(x)}{Q(x)}\)
  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: \(f(x) = \frac{x^2 + 1}{x - 1} = x + 1 + \frac{2}{x-1}\)

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

Three rational functions with dashed asymptotes. Left: 1/x, vertical asymptote x = 0 and horizontal y = 0. Middle: the average cost (1000 + 5x)/x, falling towards the horizontal asymptote y = 5. Right: (x - 2)(x + 1)/((x - 2)(x + 3)), vertical asymptote x = -3, horizontal y = 1 and a hollow dot for the hole at (2, 0.6).

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

Graph of f(x) = (x² - x - 6)/(x² - 4) in two branches, separated by the dashed vertical asymptote x = 2 and levelling off towards the dashed horizontal asymptote y = 1. A hollow dot marks the hole at (-2, 1.25); the x-intercept (3, 0) and the y-intercept (0, 1.5) are marked.

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)
  • Average cost stays above \(v\) and approaches it as \(x \to \infty\)

Let’s see an example!

Visualization of Average Cost

Two panels for fixed costs of 1000 and variable costs of 5 per unit. Left: average cost falls steeply and then levels off towards the horizontal asymptote y = 5, with AC(10) = 105 and AC(50) = 25 marked; x = 0 is a vertical asymptote. Right: the total cost line 1000 + 5x and the dashed line for average cost times quantity lie on top of each other.

Manufacturing Analysis

Work through this business scenario and then we compare

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

  1. Write the average cost function
  2. Find the horizontal asymptote and interpret it

Manufacturing Analysis II

  1. How many units bring average cost to within 5 EUR of the asymptote?
  2. Graph the function

Hint for c): The denominator is positive, so you can multiply both sides by it without changing the sign.

Guided Practice

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 \(x \to 0^+\) (positive \(x\) near 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!

Coffee Break

FSP Workshop

Problem 1: Inspection Queue

A company runs a vehicle inspection station with one inspection lane. An inspection takes 3 minutes on average, so the lane can handle at most 20 cars per hour. If on average \(x\) cars arrive per hour, the company’s planning model gives the average time a car spends at the station (waiting plus inspection) as

\[T(x) = \frac{60}{20 - x} \text{ minutes.}\]

  1. Give the domain of \(T\) in this context and interpret \(T(0)\). Determine the vertical asymptote of \(T\) and explain its meaning for the station. [4 pts.]

Inspection: Busy Hours

  1. Give the range of \(T\) on its domain. The station manager says: “If arrivals rise by 20 %, from 15 to 18 cars per hour, the time at the station also rises by about 20 %.” Assess this statement. [3 pts.]

  2. The station promises: “On average, you are back on the road within 15 minutes.” Determine all arrival rates for which the promise holds. [3 pts.]

Hint: Dividing the same positive number by a smaller positive number gives a larger result. In c), the denominator is positive, so you can multiply both sides by it without changing the sign.

Inspection: A Faster Device

A new test device cuts the inspection to 2 minutes, so the lane can handle at most 30 cars per hour. The model becomes \(T_2(x) = \frac{60}{30 - x}\).

  1. Show that \(T_2(x) = T(x - 10)\). Describe the transformation and give the new vertical asymptote. Determine, with the transformation, all arrival rates for which the 15-minute promise now holds. [4 pts.]

Inspection: Does It Pay?

  1. Each inspection brings a margin of 40 EUR (fee minus variable costs). The station is open 40 hours per week, and enough customers would come. It always works at the largest arrival rate that keeps the promise. The device costs 2,400 EUR per week. Decide whether it pays off. [2 pts.]

Wrap-Up & 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!