Combining Functions by Multiplication and Division: Algebraic and Graphical Insights

Introduction

In section 8.2 of the course, we move beyond adding and subtracting functions and explore how to combine them through multiplication and division. This article walks through the key concepts, algebraic procedures, and graphical considerations, so you can master the topic without watching the video.

Warm‑up Example

1. Given functions
2. (f(x) = 2x^2 + 3x - 5) (quadratic)
3. (g(x) = x + 4) (linear)
4. Multiplication – Define (h(x) = f(x) \cdot g(x))
5. Expand: [h(x) = (2x^2 + 3x - 5)(x + 4)]
6. Distribute each term of the quadratic across the linear factor.
7. Resulting polynomial is cubic (degree 3), as expected when a degree‑2 and a degree‑1 polynomial are multiplied.
8. Division – Define (k(x) = \frac{f(x)}{g(x)})
9. Simplify where possible; common factor (x) can be cancelled from terms that contain it.
10. The final expression remains a rational function because not all terms cancel.
11. No further simplification is possible, so the answer is left as the fraction.

Notation and Restrictions

• Multiplication can be written as (f(x)\cdot g(x)) or simply (f*g(x)).
• Division can appear as (\frac{f(x)}{g(x)}) or (f/g(x)).
• Domain restriction: Whenever a denominator becomes zero, the combined function is undefined. Set the denominator equal to zero, solve for (x), and exclude those values from the domain (often noted as a vertical asymptote in the graph).

Graphical Approach

1. Align domains – Ensure both original functions share the same (x)-values before combining.
2. Compute new (y)-values – Multiply or divide the corresponding (y)-values of the original graphs.
3. Plot the resulting points – The new graph reflects the product or quotient.
4. Observe range changes – Multiplying a cubic and a linear can produce a quartic‑like shape, limiting the range compared to the original functions.

Parity (Even/Odd) Behavior

• Multiplication
• Even × Even → Even
• Odd × Odd → Even
• Even × Odd → Odd
• Division
• Even ÷ Even → Even
• Odd ÷ Odd → Even
• Even ÷ Odd (or Odd ÷ Even) → Odd Understanding parity helps predict symmetry of the resulting function without full algebraic expansion.

Key Takeaways

• Multiplying a degree‑(m) polynomial by a degree‑(n) polynomial yields a degree‑(m+n) polynomial.
• Division creates rational functions; always check for zeros in the denominator to identify domain restrictions and possible asymptotes.
• Graphical intuition mirrors the algebraic process: align domains, compute new (y)-values, and watch for changes in range and symmetry.
• Parity rules provide quick insight into the shape (even vs. odd) of the combined function.

Next Steps

Proceed to the follow‑up video for a series of worked examples that reinforce these concepts.

Combining functions through multiplication and division expands your toolkit for building more complex models; remember to simplify algebraically, respect domain restrictions, and use parity to anticipate graph symmetry.