Math Problem Statement

Find the domain of the function g(t) = √(t² + 1).

Solution

The given function is:

g(t)=t2+1g(t) = \sqrt{t^2 + 1}

To find the domain of this function:

The domain of a square root function requires that the expression inside the square root must be non-negative (greater than or equal to zero).

Here, the expression inside the square root is:

t2+1t^2 + 1

Since t2t^2 is always non-negative (because squaring any real number results in a non-negative value), and adding 1 makes the expression strictly positive, we can conclude that:

t2+11t^2 + 1 \geq 1

Thus, t2+1t^2 + 1 is always positive for all real numbers tt.

Domain:

The function is defined for all real values of tt. Therefore, the domain of the function is:

(,)(-\infty, \infty)

Do you want more details or have any questions about this problem?

Here are some related questions you might want to explore:

  1. What is the range of the function g(t)g(t)?
  2. How do you find the domain of a rational function?
  3. What happens to the domain if there is a different expression inside the square root?
  4. Can a square root function ever have negative values in its domain?
  5. How does the graph of g(t)=t2+1g(t) = \sqrt{t^2 + 1} look like?

Tip: Remember that square root functions are defined only for non-negative values inside the root. Always check what makes the expression inside the square root valid!

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Square Roots
Domain

Formulas

The expression inside the square root must be non-negative (≥ 0)

Theorems

Properties of Square Root Functions

Suitable Grade Level

Grades 9-12