GED · Mathematical Reasoning · 6 min read
Functions and Their Representations
- Identify whether a relation given as a table, mapping diagram, set of ordered pairs, or graph represents a function.
- Evaluate functions using function notation for positive and negative inputs, and solve for inputs given a specific output.
- Distinguish between linear and nonlinear functions across equations, tables, and graphs, identifying key features like slope, intercepts, domain, and range.
Understanding Functions, Domain, and Range
A function is a specific kind of mathematical relationship where each input value has exactly one output value. Think of a function like a dependable machine: whenever you feed it a specific item, it always produces one predictable result. The input value is known as the independent variable, which is most commonly represented by the letter x. The resulting output is called the dependent variable, most commonly represented by the letter y or written as f(x).
Two essential ideas connected to functions are domain and range. The set of all valid input values for a function is called the domain. When you gather every possible number that x can be, you have the domain. The set of all resulting output values that come out of the function is called the range. In short, the domain covers all the inputs, and the range covers all the outputs.
A relation fails to be a function if a single input value corresponds to more than one output value. If you put one input into a rule and get two different answers back, that relationship is not a function. Keeping this rule in mind is the key to analyzing functions across every format you will see on test day.
Common mistakes
- Squaring a negative input without parentheses: On the TI-30XS calculator, typing -3 squared without parentheses gives -9, whereas evaluating f(-3) for f(x) = x squared requires squaring (-3) to get positive 9.
- Confusing outputs with inputs: Substituting an output value into x instead of setting the entire function expression f(x) equal to that value and solving for x.
- Assuming repeated y-values disqualify a function: Believing that a relation is not a function because two different inputs produce the same output; only a single input leading to multiple outputs violates the definition.
- Mistaking a vertical line for a function: Forgetting that a vertical line has an x-value paired with infinitely many y-values, meaning it fails the vertical line test and is never a function.
Key points
- A function is a relation where each input (independent variable x, belonging to the domain) pairs with exactly one output (dependent variable y or f(x), belonging to the range).
- To test if a relation is a function, look for repeated inputs with different outputs in tables, multiple outgoing arrows from a single input in mapping diagrams, or lines crossing twice during the vertical line test on graphs.
- Function notation f(x) names the output for input x; evaluate by substituting numbers into x, taking care to wrap negative inputs in parentheses.
- The graph of a linear function lies on a straight line and has a constant rate of change called slope (m); it can be written as f(x) = mx + b, where (0, b) is the y-intercept if 0 is in the domain.
- Nonlinear functions, such as quadratic parabolas (f(x) = ax squared + bx + c) or exponential curves (f(x) = 2^x), do not have a constant rate of change across their domain.
- Find the domain and range on a graph by tracing all x-values and y-values covered, noting any arrows, gaps, and open or closed endpoints.
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