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Chapter 0: Problem 51

find the domain of the given expression. $$ \frac{\sqrt{x+2}}{1-x} $$

Short Answer

Expert verified
The domain of the function \(\frac{\sqrt{x+2}}{1-x}\) is \([-2, 1) \cup (1, \infty)\).

Step by step solution

01

Solve Inequality for Radical

Solve the inequality \(x+2 \geq 0\) to determine for which x-values the square root will be real. This simplifies to \(x \geq -2\). This means that for the square root part of the function, x has to be greater than or equal to -2.
02

Solve Equation for Denominator

Solve the equation \(1-x \neq 0\) to find for which x-values the denominator of the function equals zero, because for those points, function will not be defined. This simplifies to \(x \neq 1\).
03

Combine Solutions

The complete domain of the function is found by combining the solutions of step 1 and step 2. Hence, domain of the given function will be \(x \geq -2\) and \(x \neq 1\). In interval notation, this is \([-2, 1) \cup (1, \infty)\).

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Key Concepts

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

Inequality Solving
In order to determine aspects of a function's domain, we often need to solve inequalities. Inequality solving involves finding values for the variable that make an inequality true. For example, in the initial step of the domain exercise, we encounter the inequality \(x + 2 \geq 0\). This inequality stems from the fact that the square root of a number is only real when that number is non-negative.

To solve this inequality:\
  • Subtract 2 from both sides to isolate \(x\).
  • The step simplifies the inequality to \(x \geq -2\).
Hence, the inequality implies that \(x\) must be greater than or equal to \(-2\). This process of solving inequalities is crucial as it helps identify values where functions like square roots remain valid and real.
Domain Restriction
When finding the domain of a function, it's essential to determine where the function is defined and valid. This involves checking restrictions implied by various parts of the function. With rational functions, domain restrictions arise from the denominators, which cannot be zero as division by zero is undefined.

In the example given, the denominator is \(1-x\). Solve the equation \(1-x eq 0\) to see which values of \(x\) cause the denominator to become zero.
  • Rearrange the equation to get \(x eq 1\).
This restriction means \(x\) cannot equal 1, since division by 0 is not possible. By understanding domain restrictions, we avoid undefined points and ensure a real function.
Interval Notation
Once you've identified the domain restrictions from inequalities and restrictions, you can express the domain using interval notation. Interval notation provides a clear, compact way to describe all permissible values of \(x\).

In the exercise, the domain is adapted by:
  • Combining the inequality solution \(x \geq -2\) with the restriction \(x eq 1\).
This leads to the domain being expressed as \([-2, 1) \cup (1, \infty)\).
  • \([-2, 1)\) denotes all \(x\) from -2 to just before 1, including -2 and excluding 1.
  • \((1, \infty)\) covers all \(x\) greater than 1.
With these intervals, we cover all valid \(x\)-values for the function, neatly avoiding any undefined points.

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Most popular questions from this chapter

Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. $$ x^{3}+2 x^{2}-5 x-6 $$

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