Math Problem Statement

Find domain and range

Solution

The expression in the image is:

y3(x9)=1y^3(x - 9) = 1

Finding the Domain

To find the domain of the function, we need to determine for which values of xx and yy the expression is defined.

  • The expression y3(x9)=1y^3(x - 9) = 1 implies that the expression is defined for all xx and yy such that x90x - 9 \neq 0 (since y30y^3 \neq 0 for any real number yy).
  • Therefore, x9x \neq 9.

Thus, the domain of the function is all real numbers except x=9x = 9: Domain: xR,  x9.\text{Domain: } x \in \mathbb{R}, \; x \neq 9.

Finding the Range

To find the range, we solve the equation for yy:

y3=1x9y^3 = \frac{1}{x - 9}

Since x9x \neq 9, x9x - 9 can take any non-zero real value. Therefore, 1x9\frac{1}{x - 9} can also take any non-zero real value.

  • Since y3=1x9y^3 = \frac{1}{x - 9}, y3y^3 can be any non-zero real number.
  • yy itself can therefore be any non-zero real number.

Thus, the range of the function is all real numbers except y=0y = 0: Range: yR,  y0.\text{Range: } y \in \mathbb{R}, \; y \neq 0.

Would you like more details, or do you have any questions?

Related Questions:

  1. What happens to the graph of the function as xx approaches 9?
  2. Can we rewrite this expression in terms of inverse functions?
  3. How would you describe the behavior of yy as xx becomes very large or very small?
  4. What are the asymptotes for this function, if any?
  5. How would you plot this function on a coordinate plane?

Tip:

When determining the domain and range of a function, always consider the values that make the function undefined, such as division by zero or taking the square root of a negative number in the real number system.

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Math Problem Analysis

Mathematical Concepts

Functions
Domain and Range

Formulas

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Theorems

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Suitable Grade Level

High School