Functions and Graphs - Examples & Applications

This section provides step-by-step examples on how to analyze and manipulate functions. You will learn to find domains, evaluate composite functions, determine symmetry algebraically, and apply graphical transformations.

Function Notation and Tests

Function notation f(x)f(x) replaces yy and explicitly shows the independent variable. The Vertical Line Test determines if a graph represents a function (one yy for every xx).

Example

Case Study 1: Is it a Function? Determine whether the following relations represent a function:
  1. The set of points: {(1,2),(3,4),(5,6),(1,8)}\lbrace(1, 2), (3, 4), (5, 6), (1, 8)\rbrace
  2. The equation: x2+y2=25x^2 + y^2 = 25

Step-by-Step Solution

0 of 2 Steps Completed
1

Example

Case Study 2: Evaluating Functions and Expressions Given f(x)=2x2−3x+1f(x) = 2x^2 - 3x + 1, evaluate the following:
  1. f(4)f(4)
  2. f(−2)f(-2)
  3. f(x+h)f(x + h)

Step-by-Step Solution

0 of 3 Steps Completed
1

Determining Domain

The domain of a function is the set of all valid inputs (xx-values). The two most common restrictions are division by zero and taking the even root of a negative number.

Example

Example 1: Rational Function Domain (Basic) Find the domain of f(x)=x+5x2−9f(x) = \frac{x + 5}{x^2 - 9}.

Step-by-Step Solution

0 of 3 Steps Completed
1

Example

Example 2: Radical Function Domain (Intermediate) Find the domain of g(x)=2x−8g(x) = \sqrt{2x - 8}.

Step-by-Step Solution

0 of 3 Steps Completed
1

Example

Example 3: Combined Restrictions (Advanced) Find the domain of h(x)=15−xh(x) = \frac{1}{\sqrt{5 - x}}.

Step-by-Step Solution

0 of 4 Steps Completed
1

Symmetry: Even and Odd Functions

Symmetry can be proven algebraically. An even function is symmetric about the y-axis (f(−x)=f(x)f(-x) = f(x)). An odd function is symmetric about the origin (f(−x)=−f(x)f(-x) = -f(x)).

Example

Example 1: Testing for Even Symmetry Determine algebraically if f(x)=3x4−2x2+5f(x) = 3x^4 - 2x^2 + 5 is even, odd, or neither.

Step-by-Step Solution

0 of 3 Steps Completed
1

Example

Example 2: Testing for Odd Symmetry Determine algebraically if g(x)=2x3−xg(x) = 2x^3 - x is even, odd, or neither.

Step-by-Step Solution

0 of 4 Steps Completed
1

Types of Functions

Classifying functions helps predict their graphs and identify key features.

Example

Example 1: Identifying Function Types (Basic) Classify: 1) f(x)=3x2−2x+1f(x) = 3x^2 - 2x + 1, 2) g(x)=2x−1x+3g(x) = \frac{2x - 1}{x + 3}, 3) h(x)=5(2x)h(x) = 5(2^x).

Step-by-Step Solution

0 of 3 Steps Completed
1

Function Composition

Composition involves substituting an entire function into another function, denoted as (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)).

Example

Example 1: Composing Functions (Intermediate) Given f(x)=x2+3f(x) = x^2 + 3 and g(x)=2x−1g(x) = 2x - 1, find (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x).

Step-by-Step Solution

0 of 4 Steps Completed
1

Function Transformations

Transformations alter the shape or position of a parent function's graph.

Example

Example 1: Analyzing Transformations (Intermediate) Describe the transformations applied to f(x)=x2f(x) = x^2 to obtain g(x)=−2(x−3)2+4g(x) = -2(x - 3)^2 + 4.

Step-by-Step Solution

0 of 3 Steps Completed
1

Direct, Inverse, and Joint Variation

Variation models describe how variables relate proportionally using a constant kk.

Example

Example 1: Inverse Variation Problem (Advanced) Time tt varies inversely with speed vv. If it takes 44 hours at 6060 mph, how long will it take at 8080 mph?

Step-by-Step Solution

0 of 3 Steps Completed
1
PreviousFunctions and Graphs - Theory & Concepts
Quiz Me
NextLogarithms - Theory & Concepts