Find the domain of the function. (Enter your answer using interval notation.)
[tex]g(x)=\frac{\sqrt{2+x}}{6-x}[/tex]
Answer (1)
The expression inside the square root must be non-negative: 2 + x ≥ 0 , which means x ≥ − 2 .
The denominator cannot be zero: 6 − x = 0 , which means x = 6 .
Combining these restrictions, the domain is [ − 2 , 6 ) ∪ ( 6 , ∞ ) .
Therefore, the domain of the function is [ − 2 , 6 ) ∪ ( 6 , ∞ ) .
Explanation
Analyzing the Problem We are asked to find the domain of the function g ( x ) = 6 − x 2 + x . The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have two restrictions to consider:
The expression inside the square root must be non-negative, i.e., 2 + x ≥ 0 .
The denominator cannot be zero, i.e., 6 − x = 0 .
Solving the Inequality First, let's solve the inequality 2 + x ≥ 0 . Subtracting 2 from both sides, we get: x ≥ − 2 This means that x must be greater than or equal to -2 for the square root to be defined.
Finding the Restriction from the Denominator Next, let's find the value of x that makes the denominator zero. We have 6 − x = 0 . Adding x to both sides, we get: x = 6 This means that x cannot be equal to 6, because the denominator would be zero, and the function would be undefined.
Combining the Restrictions Now, we need to combine these two restrictions. We know that x ≥ − 2 and x = 6 . In interval notation, this can be written as: [ − 2 , 6 ) ∪ ( 6 , ∞ ) This means that x can be any value from -2 up to (but not including) 6, and any value greater than 6.
Examples
Understanding the domain of a function is crucial in many real-world applications. For instance, if g ( x ) represents the amount of drug in a patient's bloodstream x hours after administration, the domain tells us the valid time frame for which the model is applicable. We can't have negative time, and there might be a point where the model breaks down due to drug interactions or elimination. Similarly, in physics, if g ( x ) represents the height of a projectile, the domain would specify the valid range of horizontal distances for which the height is defined.
