Math Problem Statement

The user uploaded an image containing several functions and requested their analysis or assistance.

Solution

I see that you've uploaded an image containing several functions. Here's the breakdown of the functions provided in the image:

  1. f(x)=x26x+8f(x) = \sqrt{x^2 - 6x + 8}
  2. f(x)=log3(4x2x1)f(x) = \log_3 \left( \frac{4 - x^2}{x - 1} \right)
  3. f(x)=25x2log21(x+3)f(x) = \sqrt{25 - x^2} - \log_{21}(x + 3)
  4. f(x)=x2x+2+log4(sinx)f(x) = \sqrt{-x^2 - x + 2} + \log_4 (\sin x)

Would you like help analyzing or solving any of these functions? Let me know which specific part you'd like assistance with, or if you have questions about them.

Some related questions you may find useful:

  1. How do I determine the domain of these functions?
  2. What is the range of the function f(x)=x26x+8f(x) = \sqrt{x^2 - 6x + 8}?
  3. How do logarithms with different bases work, such as log3\log_3 and log21\log_{21}?
  4. How do I simplify the expression 25x2log21(x+3)\sqrt{25 - x^2} - \log_{21}(x + 3)?
  5. What is the significance of the function f(x)=x2x+2f(x) = \sqrt{-x^2 - x + 2} being under the square root?

Tip: When dealing with functions involving square roots and logarithms, always start by identifying the domain restrictions to avoid undefined expressions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Logarithms
Functions
Domain and Range Analysis

Formulas

Square root function \( \sqrt{x} \)
Logarithmic properties \( \log_b(a) \)
Quadratic equations \( ax^2 + bx + c \)

Theorems

Properties of square roots
Domain restrictions for logarithms
Quadratic discriminant analysis

Suitable Grade Level

Grades 9-12