Functions: Definitions, Operations, and Applications

Functions & Graphs

Definition and Identification of Functions

A function is a relation in which each input (usually x) has exactly one output (usually y). Not all equations represent functions.

• Vertical Line Test: If any vertical line crosses a graph more than once, the relation is not a function.

• Example: $x^3 + y = 2$ can be solved for y as $y = 2 - x^3$, which is a function.

• Non-Example: $x^2 + y^2 = 4$ is a circle; for some x-values, there are two y-values, so it is not a function.

Function Notation and Evaluation

Functions are often written as $f(x)$, which means the output when x is the input. To evaluate a function, substitute the given value for x.

• Example: If $f(x) = 2x^2 + 3x$, then $f(3) = 2(3)^2 + 3(3) = 18 + 9 = 27$.

Operations with Functions

Functions can be added, subtracted, multiplied, or divided. The result is a new function.

• Addition: $(f+g)(x) = f(x) + g(x)$

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

• Multiplication: $(fg)(x) = f(x) imes g(x)$

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

• Example: If $f(x) = x^2 + 9$, $g(x) = 3x + 5$, then

  • $(f+g)(2) = f(2) + g(2) = 4 + 9 + 6 + 5 = 24$

  • $(f-g)(5) = f(5) - g(5) = 25 + 9 - (15 + 5) = 34 - 20 = 14$

  • $(fg)(-2) = f(-2) imes g(-2) = (4 + 9) imes (-6 + 5) = 13 imes (-1) = -13$

  • $(\frac{f}{g})(1) = \frac{f(1)}{g(1)} = \frac{1 + 9}{3 + 5} = \frac{10}{8} = \frac{5}{4}$

Difference Quotient

The difference quotient is a formula used to compute the average rate of change of a function over an interval. It is foundational for calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$, where $h \neq 0$

• Example: For $f(x) = 2x^2 + 3x$, $f(x+h) = 2(x+h)^2 + 3(x+h) = 2(x^2 + 2xh + h^2) + 3x + 3h = 2x^2 + 4xh + 2h^2 + 3x + 3h$ $f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 + 3x + 3h) - (2x^2 + 3x) = 4xh + 2h^2 + 3h$ $\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 + 3h}{h} = 4x + 2h + 3$

Factoring and Expanding Expressions

Factoring and expanding are essential algebraic skills for manipulating functions.

• Expanding: $(a+b)^2 = a^2 + 2ab + b^2$

• Example: $(x+h)^2 = x^2 + 2xh + h^2$

Rational Functions: Domain and Intercepts

A rational function is a function of the form $g(x) = \frac{x+1}{x+2}$. The domain is all real numbers except where the denominator is zero.

• Domain: $x+2 \neq 0 \implies x \neq -2$ Domain: $(-\infty, -2) \cup (-2, \infty)$

• Finding Points on the Graph: To check if a point $(a, b)$ is on the graph, substitute $x = a$ and see if $g(a) = b$.

• Solving for x: If $g(x) = 2$, solve $\frac{x+1}{x+2} = 2 \implies x+1 = 2(x+2) \implies x+1 = 2x+4 \implies x = -3$

• Intercepts:

  • x-intercept: Set $g(x) = 0 \implies x+1 = 0 \implies x = -1$

  • y-intercept: Set $x = 0 \implies g(0) = \frac{1}{2}$

Summary Table: Function Operations

Operation	Formula	Example
Addition	$(f+g)(x) = f(x) + g(x)$	$(f+g)(2) = 24$
Subtraction	$(f-g)(x) = f(x) - g(x)$	$(f-g)(5) = 14$
Multiplication	$(fg)(x) = f(x)g(x)$	$(fg)(-2) = -13$
Division	$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$	$(\frac{f}{g})(1) = \frac{5}{4}$

Additional info:

• These notes cover the basics of functions, including identification, evaluation, operations, the difference quotient, and rational functions. Mastery of these concepts is essential for further study in Precalculus and Calculus.