Operations with Functions: Addition, Subtraction, Multiplication, and Division

Operations with Functions

Introduction

In College Algebra, understanding how to perform operations with functions is essential. These operations include addition, subtraction, multiplication, and division of functions. Each operation follows specific rules, and determining the domain of the resulting function is a crucial step.

Adding and Subtracting Functions

Definition and Process

• Addition and subtraction of functions are performed by combining like terms, similar to how polynomials are added or subtracted.

• If f(x) and g(x) are two functions, then:

  • Sum: $(f + g)(x) = f(x) + g(x)$

  • Difference: $(f - g)(x) = f(x) - g(x)$

• The domain of $(f + g)$ or $(f - g)$ is the set of all x-values common to the domains of both f and g.

Example

• Given $f(x) = x^2 + 4$ and $g(x) = 5x + 7$:

  • $(f + g)(x) = (x^2 + 4) + (5x + 7) = x^2 + 5x + 11$

  • $(f - g)(x) = (x^2 + 4) - (5x + 7) = x^2 - 5x - 3$

• Domain: Since both are polynomials, the domain is all real numbers, $(-cfty, 5cfty)$.

Example with Restricted Domain

• Given $f(x) = \sqrt{x + 4} + 30$ and $g(x) = \sqrt{x + 4} - 2x + 35$:

  • $(f + g)(x) = (\sqrt{x + 4} + 30) + (\sqrt{x + 4} - 2x + 35) = 2\sqrt{x + 4} - 2x + 65$

  • $(f - g)(x) = (\sqrt{x + 4} + 30) - (\sqrt{x + 4} - 2x + 35) = 2x - 5$

• Domain: $x + 4 \geq 0 \implies x \geq -4$ (since both functions involve $\sqrt{x + 4}$).

Multiplying and Dividing Functions

Definition and Process

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$

• The domain of $(f \cdot g)$ is the set of all x-values common to the domains of both f and g.

• The domain of $\left(\frac{f}{g}\right)$ is the set of all x-values common to the domains of both f and g, excluding any x for which $g(x) = 0$.

Example

• Given $f(x) = \sqrt{x}$ and $g(x) = 3x - 6$:

  • Domain of $f(x)$: $x \geq 0$

  • Domain of $g(x)$: all real numbers

  • Domain of $f \cdot g$: $x \geq 0$

  • Domain of $\frac{f}{g}$: $x \geq 0$ and $g(x) \neq 0 \implies 3x - 6 \neq 0 \implies x \neq 2$

Example with Polynomials

• Given $h(x) = 2x^2 - 4$ and $k(x) = x^2 + 2$:

  • $(h \cdot k)(x) = (2x^2 - 4)(x^2 + 2)$

  • $(h / k)(x) = \frac{2x^2 - 4}{x^2 + 2}$

  • Domain: Both are defined for all real numbers, but for division, $x^2 + 2 \neq 0$ (which is always true for real x), so the domain is all real numbers.

Determining Domains of Combined Functions

Key Points

• The domain of a combined function is the intersection of the domains of the original functions.

• For division, exclude any x-values that make the denominator zero.

• For functions involving square roots, ensure the radicand is non-negative.

Summary Table: Operations with Functions

Operation	Formula	Domain
Addition	$(f + g)(x) = f(x) + g(x)$	Common to domains of f and g
Subtraction	$(f - g)(x) = f(x) - g(x)$	Common to domains of f and g
Multiplication	$(f \cdot g)(x) = f(x) \cdot g(x)$	Common to domains of f and g
Division	$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$	Common to domains of f and g, $g(x) \neq 0$

Practice Problems

• Given $L(x) = x - 2$ and $M(x) = x^3$, calculate $\frac{L}{M}(5)$:

  • $L(5) = 5 - 2 = 3$

  • $M(5) = 125$

  • $\frac{L}{M}(5) = \frac{3}{125}$

• Given $h(x) = 2x^2 - 4$ and $k(x) = x^2 + 2$, find $h - k$:

  • $(h - k)(x) = (2x^2 - 4) - (x^2 + 2) = x^2 - 6$

Additional info: Always determine the domain restrictions before simplifying the functions. For radical and rational functions, pay special attention to values that make the radicand negative or the denominator zero.