Function Operations: Addition, Subtraction, Multiplication, and Division of Functions

Function Operations

Adding & Subtracting Functions

In precalculus, functions can be added or subtracted in a manner similar to polynomials. The resulting function's domain is the intersection of the domains of the original functions.

• Definition: If $f(x)$ and $g(x)$ are functions, then their sum and difference are defined as: $$(f+g)(x) = f(x) + g(x)$$(f-g)(x) = f(x) - g(x)$$

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

• Example: If $f(x) = x^2 + 4$ and $g(x) = 5x + 7$, then: $$(f+g)(x) = x^2 + 4 + 5x + 7 = x^2 + 5x + 11$$

Additional info: Always check for domain restrictions, especially when functions involve roots or denominators.

Multiplying & Dividing Functions

Functions can also be multiplied or divided. The domain of the resulting function is determined by the intersection of the domains of the original functions, with additional restrictions for division.

• Definition:

  • Product: $$(f imes g)(x) = f(x) imes g(x)$$

  • Quotient: $$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$$, where$g(x) \neq 0$

• Domain:

  • For multiplication: The domain is the set of $x$ values common to both $f$ and $g$.

  • For division: The domain is the set of $x$ values common to both $f$ and $g$, excluding values where $g(x) = 0$.

• Example: If $f(x) = \sqrt{x}$$and$, then: $$(f \times g)(x) = \sqrt{x} \times (3x - 6)$$Domain:$x \geq 0$ and $x \neq 2$

Additional info: For division, always exclude points where the denominator is zero.

Determining Domains in Function Operations

When performing operations on functions, it is essential to determine the domain of the resulting function before simplifying.

• Key Steps:

  1. Identify the domains of each function involved.

  2. For addition, subtraction, and multiplication, use the intersection of the domains.

  3. For division, use the intersection and exclude points where the denominator is zero.

• Example: Given $f(x) = x^2 + 4$ and $g(x) = \sqrt{x - 2}$, the domain of $g(x)$ is $x \geq 2$. The domain of $f(x)$ is all real numbers. The domain of $f + g$ is $x \geq 2$.

Practice Problems

• Example 1: Given $f(x) = x^2 + 4$ and $g(x) = \sqrt{x - 2}$, find $(f + g)(x)$ and its domain. $$(f + g)(x) = x^2 + 4 + \sqrt{x - 2}$$Domain:$x \geq 2$

• Example 2: Given $h(x) = 2x^2 + 4$ and $k(x) = x^2$, find $h - k$ and its domain. $$(h - k)(x) = 2x^2 + 4 - x^2 = x^2 + 4$$ Domain: All real numbers

• Example 3: Given $l(x) = x - 2$ and $M(x) = x^2$, calculate $M(5)$. $$M(5) = 5^2 = 25$$

Summary Table: Function Operations and Domains

Operation	Formula	Domain Rule
Addition	$(f+g)(x) = f(x) + g(x)$	Intersection of domains of $f$ and $g$
Subtraction	$(f-g)(x) = f(x) - g(x)$	Intersection of domains of $f$ and $g$
Multiplication	$(f \times g)(x) = f(x) \times g(x)$	Intersection of domains of $f$ and $g$
Division	$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$	Intersection of domains of $f$ and $g$, $g(x) \neq 0$

Additional info: These operations are foundational for understanding more advanced function manipulations in precalculus and calculus.