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Chapter 3: Problem 94

Write the domain of the function in interval notation. $$ g(t)=\sqrt{1-t^{2}} $$

Short Answer

Expert verified
The domain is \([ -1, 1 ])\.

Step by step solution

01

Identify the Condition for the Radicand

A square root function is defined only when the radicand (the expression inside the square root) is greater than or equal to zero. Set the expression inside the square root greater than or equal to zero: \[ 1 - t^2 \geq 0 \]
02

Solve the Inequality

Rearrange the inequality to find the possible values of t. Solve: \[ 1 \geq t^2 \] Taking square roots on both sides, we get: \[ -1 \leq t \leq 1 \]
03

Express in Interval Notation

The values of t range from -1 to 1, inclusive. In interval notation, write the domain as:\[ [-1, 1] \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a way to describe a set of numbers along a number line. This approach helps us specify the domain or range of a function in a concise manner. To use interval notation, we use brackets and parentheses:
If the endpoints are included, we use square brackets \([a, b]\). This means \('a' and 'b' are included in the interval\).
If the endpoints are not included, we use parentheses \((a, b)\). This implies \('a' and 'b' are not included\).
For example, \([2, 5)\) includes 2 but not 5. Make sure to correctly identify if the endpoints are inclusive or exclusive to accurately represent the domain or range of a function.
Inequality Solving
Inequality solving involves finding the set of values for a variable that satisfies a given inequality. To solve inequalities:
  • Isolate the variable on one side by performing arithmetic operations.
  • Use inverse operations to simplify.
  • Be cautious when multiplying or dividing by negative numbers; the inequality sign flips its direction in such cases.
Consider the inequality \( 1-t^2 \ge 0 \):
- Rearrange to \( 1 \ge t^2 \).
- Solve it into \( t \le 1 \) and \( t \ge -1 \). Combining them, we get \( -1 \le t \le 1 \). Inequality solving helps identify valid values for variables, crucial for defining domains in functions.
Square Root Function
The square root function involves taking the root of a number or expression. The domain of a square root function includes only values for which the radicand (inside the root) is non-negative. For example, in \[ g(t)=\root1-t^2 \], the radicand \( 1-t^2 \) must be \( \ge 0 \). Steps:
  • Find the condition where the radicand is non-negative: \( 1 - t^2 \ge 0 \).
  • Rearrange and solve for \( t \): \( -1 \le t \le 1 \).
  • Convey the result in interval notation: \( \[ -1, 1\] \).
Understanding these conditions ensures correctly finding and expressing the function's domain.

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Most popular questions from this chapter

Suppose that \(y\) varies jointly as \(x^{4}\) and \(w\). If \(x\) is replaced by \(\frac{1}{4} x\) and \(w\) is replaced by \(4 w,\) what is the effect on \(y ?\)

Given the inequality, \(0.552 x^{3}+4.13 x^{2}-1.84 x-3.5<6.7\) a. Write the inequality in the form \(f(x)<0\). b. Graph \(y=f(x)\) on a suitable viewing window. c. Use the Zero feature to approximate the real zeros of \(f(x)\). Round to 1 decimal place. d. Use the graph to approximate the solution set for the inequality \(f(x)<0\)

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