Math Problem Statement

Answer

The domain of the function f(x) is (-∞, 20] and the range is [-20, 8]. Using interval notation, find the domain and range of g(x) = f(x - 6).

Solution

We are given that:

  • The domain of f(x)f(x) is (,20](-\infty, 20]
  • The range of f(x)f(x) is [20,8][-20, 8]

We need to find the domain and range of the function g(x)=f(x6)g(x) = f(x - 6).

1. Finding the Domain of g(x)g(x):

The transformation g(x)=f(x6)g(x) = f(x - 6) represents a horizontal shift of the function f(x)f(x) to the right by 6 units.

  • If the domain of f(x)f(x) is (,20](-\infty, 20], after shifting the graph to the right by 6 units, the new domain will be shifted accordingly.
  • Therefore, the domain of g(x)g(x) is (+6,20+6]=(,26](-\infty + 6, 20 + 6] = (-\infty, 26].

2. Finding the Range of g(x)g(x):

A horizontal shift does not affect the range of the function. Therefore, the range of g(x)g(x) remains the same as the range of f(x)f(x).

  • The range of g(x)g(x) is still [20,8][-20, 8].

Final Answer:

  • Domain of g(x)g(x): (,26](-\infty, 26]
  • Range of g(x)g(x): [20,8][-20, 8]

Would you like more details or clarifications?

Related Questions:

  1. What happens to the domain when shifting the graph left by 4 units?
  2. How would the range change if we applied a vertical shift instead?
  3. What transformation would cause the range to change but not the domain?
  4. How do we handle vertical stretches or compressions when determining range?
  5. Can you find the inverse of the function f(x)f(x) and determine its domain and range?

Tip: When a function is shifted horizontally, the domain changes, but the range remains unaffected.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Functions
Transformations
Domain
Range

Formulas

-

Theorems

-

Suitable Grade Level

Grades 9-12