Quiz 2-1 Characteristics of Functions Part 1
Quiz 2-1 Characteristics of Functions Part 1 tests the foundational skills needed to recognize and describe functions using equations, tables, graphs, and mapping diagrams. The central idea is simple: a function assigns exactly one output to every input in its domain. Understanding this rule makes it possible to identify domains, ranges, intercepts, zeros, symmetry, and other important characteristics.
A solid understanding of functions is essential in algebra, calculus, physics, economics, computer science, and many other fields. These subjects often require students to interpret formulas, analyze how values change, and determine what a graph represents. This article provides a concise review followed by a comprehensive practice quiz with a complete answer key.
Core Concepts About Functions
A relation is a collection of ordered pairs of input and output values. A relation becomes a function when every input is paired with exactly one output.
To give you an idea, the relation
[ y=2x+3 ]
is a function because every value of (x) produces one and only one value of (y). In contrast,
[ x^2+y^2=9 ]
is not a function when considered as a relation between (x) and (y), because some inputs correspond to two different outputs. A circle passes the vertical line test only at certain points, so it cannot represent (y) as a function of (x).
Domain and Range
The domain is the set of all valid input values. The range is the set of all output values produced by the function.
In function notation, (f(x)) means “the output of function (f) when the input is (x).” Here's one way to look at it: if
[ f(x)=3x-2, ]
then (f(4)=3(4)-2=10).
The domain may be restricted by:
- Square roots of negative numbers
- Division by zero
- Logarithms of nonpositive numbers
- Contextual limitations, such as time or available quantities
The range may be determined from a table, graph, mapping diagram, or algebraic analysis Most people skip this — try not to..
Intercepts and Zeros
An (x)-intercept occurs where the graph crosses or touches the (x)-axis. At an (x)-intercept, (y=0). An (x)-zero is therefore an input for which
[ f(x)=0. ]
A (y)-intercept occurs where the graph crosses or touches the (y)-axis. At the (y)-intercept, (x=0), so its coordinates are ((0,f(0))).
Symmetry
A graph is even when
[ f(-x)=f(x). ]
Even functions are symmetric across the (y)-axis. The graph of (f(x)=x^2) is an example It's one of those things that adds up. That's the whole idea..
A graph is odd when
[ f(-x)=-f(x). ]
Odd functions are symmetric around the origin. The graph of (f(x)=x^3) is an example And that's really what it comes down to..
Increasing and Decreasing Behavior
A function is:
- Increasing on an interval when its output values rise as the input values move from left to right.
- Decreasing on an interval when its output values fall as the input values move from left to right.
For (f(x)=x^2), the function decreases on ((-\infty,0)) and increases on ((0,\infty)).
Extrema and End Behavior
An absolute maximum is the greatest output value in the entire domain. An absolute minimum is the smallest output value in the entire domain.
Local extrema are turning points within a broader interval. To give you an idea, (f(x)=
