How to Understand Functions: Inputs, Outputs and Examples

A function gives each allowed input exactly one output. Different inputs may share an output. To decide whether a table or set of ordered pairs is a function, look for the same input paired with different outputs. In function notation, \(f(3)\) means the output for input 3, not multiplication.

Recognize a function before using its formula

Consider the pairs \((1,5),(2,5),(3,7)\). They form a function: each input has just one output, even though 1 and 2 both give 5. Adding \((1,8)\) would break the function rule because input 1 would now give both 5 and 8.

For \(f(x)=2x+1\), substitute the input wherever \(x\) appears: \(f(3)=2(3)+1=7\). The domain is the set of allowed inputs; the range is the set of outputs actually produced. On a graph, the vertical line test checks whether any one input has more than one output.

A roadmap from the basics to later topics

Start with definition, notation, domain, range, and the worked examples below. Composition and inverses come next; limits, series, and transforms are later topics, not prerequisites for understanding a function.

Step 1: Grasp the Fundamental Definition

• A function is a special type of relation where each input (from the domain) corresponds to exactly one output (in the range).

Step 2: Differentiate Between Relations and Functions

• Not all relations are functions. A vertical line test can determine this visually: if a vertical line intersects a graph at more than one point, it’s not a function.

Step 3: Explore Various Representations

• Functions can be presented in multiple formats: algebraic equations, graphs, tables, or even verbal descriptions. Familiarize yourself with these variations.

Step 4: Work through Function Notation

• Understand the notation (f(x)) where (f) denotes the function, and (x) is the input. The entire expression represents the output for the given (x).

Step 5: Understand the Domain and Range in-depth

• As discussed previously, the domain comprises possible input values, while the range consists of resultant output values. For complex functions, determining these might involve solving inequalities or analyzing asymptotic behavior.

Step 6: Explore Different Types of Functions

• Linear Functions: Represented by straight lines.
• Quadratic Functions: Parabolic in nature.
• Exponential and Logarithmic Functions: Deal with growth and decay.
• Trigonometric Functions: Originating from circle geometry.
• Rational, Radical, and Polynomial Functions: Various algebraic forms with their peculiarities.
• Piecewise Functions: Defined in pieces, each valid in a specific interval.
• Implicit Relations: Equations not solved for one variable may define a function on a suitable domain, but not every implicit relation is a function.

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Step 7: Jump into Composite and Inverse Functions

• Composite Functions: Formed by applying one function after another ((f(g(x)))).
• Inverse Functions: Reverse the input-output relationship. An inverse is a function only when the original function is one-to-one on its domain; a domain restriction may be needed. Inverse notation does not mean a reciprocal.

Step 8: Examine Function Transformations

• Familiarize yourself with how functions change with transformations: translations (shifts), dilations (stretches/shrinks), and reflections.

Step 9: Study Limit Behavior and Continuity

• Understand the concept of limits, and how functions behave as they approach certain values.
• Explore what it means for a function to be continuous or discontinuous at a point.

Step 10: Operate with Functions

• Learn to add, subtract, multiply, and divide functions.
• Understand the resultant domain restrictions.

Step 11: Experiment with Real-world Applications

• Recognize how functions model various real-world phenomena, from population growth to sound waves.

Step 12: Engage in Advanced Topics (for the avid learner)

• Taylor and Maclaurin Series: Approximating functions with polynomials.
• Fourier Series: Decomposing functions into sinusoidal components.
• Laplace and Z-transforms: Used in engineering and control systems.

Step 13: Continual Reflection and Exploration

• Revisit concepts, practice with various problems, and seek deeper understanding.
• Use tools like graphing calculators or software (e.g., Desmos, MATLAB) to visualize and experiment.

Worked review and existing practice

Check ordered pairs

Example 1:

Given the set of ordered pairs (S=){((4,6),(5,7),(6,8),(4,9))}, does (S) represent a function?

Solution:

Notice that the input value (4) corresponds to both (6) and (9). Since the same input has two different outputs, (S) does not represent a function.

Example 2:

Given the set of ordered pairs (T=){((−2,0),(0,−1),(2,1),(3,2))}, does (T) represent a function?

Solution:

Every input value in (T) corresponds to exactly one output. Therefore, (T) does represent a function.

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