Graphs and Coordinates: Functions, Domains, and Intercepts

Graphs and Coordinates

Functions and Relations

Understanding the distinction between relations and functions is fundamental in precalculus. A relation is any set of ordered pairs, while a function is a special type of relation where each input (domain value) corresponds to exactly one output (range value).

• Definition: A function from set A to set B assigns each element of A to exactly one element of B.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: For the relation {(1, 7), (2, 8), (3, 7)}, the domain is {1, 2, 3} and the range is {7, 8}.

Evaluating Functions

To evaluate a function, substitute the given value of the independent variable into the function's formula.

• Example: If $g(x) = x^3 - 3$, then $g(0) = 0^3 - 3 = -3$.

• Application: Function evaluation is used to determine specific outputs for given inputs, which is essential for graphing and analysis.

Difference Quotient

The difference quotient is a formula that gives the average rate of change of a function over an interval. It is foundational for understanding derivatives in calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: If $f(x) = x^2 + x + 1$, then:

  • $f(x+h) = (x+h)^2 + (x+h) + 1 = x^2 + 2xh + h^2 + x + h + 1$

  • $f(x+h) - f(x) = (x^2 + 2xh + h^2 + x + h + 1) - (x^2 + x + 1) = 2xh + h^2 + h$

  • So, $\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + h}{h} = 2x + h + 1$

Evaluating and Simplifying Functions

When given a function and a specific input, substitute and simplify as required.

• Example: If $h(x) = x^4 + 5x^2 - 11$, then $h(3x) = (3x)^4 + 5(3x)^2 - 11 = 81x^4 + 45x^2 - 11$

Equations Defining Functions

To determine if an equation defines y as a function of x, solve for y and check if each x-value yields only one y-value.

• Example: For $[y] - 2y = 6$, if [y] denotes the greatest integer function, further analysis is needed. If it is a linear equation, it can be solved for y, indicating y is a function of x.

• Key Point: If for every x there is only one y, then y is a function of x.

Intercepts of Graphs

Intercepts are points where a graph crosses the axes. The x-intercept is where the graph crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0).

• Finding x-intercepts: Set y = 0 and solve for x.

• Finding y-intercepts: Set x = 0 and solve for y.

• Example: If a line crosses the x-axis at (2, 0) and the y-axis at (0, -6), these are the intercepts.

Summary Table: Key Concepts

Concept	Definition	Example
Function	Each input has exactly one output	{(1, 2), (2, 3)}
Domain	Set of all possible x-values	{1, 2, 3}
Range	Set of all possible y-values	{2, 3}
Difference Quotient	Average rate of change	$\frac{f(x+h) - f(x)}{h}$
Intercepts	Where graph crosses axes	(2, 0), (0, -6)