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.

Example 1: Function Notation and Tests (Is it a Function?)

Problem: 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

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Example 2: Combined Restrictions (Advanced Domain)

Problem: Find the domain of h(x)=15−xh(x) = \frac{1}{\sqrt{5 - x}}.

Solution:

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Example 3: Composing Functions (Intermediate)

Problem: 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).

Solution:

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Example 4: Analyzing Transformations (Intermediate)

Problem: 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.

Solution:

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Example 5: Inverse Variation Problem (Advanced)

Problem: Time tt varies inversely with speed vv. If it takes 44 hours at 6060 mph, how long will it take at 8080 mph?

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