Which of the following describes the domain of the piecewise function:
[Format: g(x) = { x²+3x / x²+x-6 for x<3, log2(x+5) for x≥3 }]
A. (-∞,∞)
B. (0, 2) ∪ (2,∞)
C. (0, 2) ∪ (2, 3) ∪ (3,∞)
D. (-∞,-3)∪(-3, 2) ∪ (2, ∞)
Answer (2)
Determine the domain of the first part of the piecewise function x 2 + x − 6 x 2 + 3 x for x < 3 , excluding x = − 3 and x = 2 .
Determine the domain of the second part of the piecewise function lo g 2 ( x + 5 ) for x ≥ 3 , which requires -5"> x > − 5 .
Combine the domains of both parts, resulting in ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) ∪ [ 3 , ∞ ) .
Simplify the combined domain to get the final answer: ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .
( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ )
Explanation
Analyze the piecewise function First, let's analyze the given piecewise function:
g ( x ) = { x 2 + x − 6 x 2 + 3 x lo g 2 ( x + 5 ) for x < 3 for x ≥ 3
We need to find the domain of this function, which means we need to find all possible values of x for which the function is defined.
Analyze the first part of the function For the first part of the function, g ( x ) = x 2 + x − 6 x 2 + 3 x for x < 3 , we need to consider the denominator. The denominator cannot be equal to zero, so we need to find the values of x for which x 2 + x − 6 = 0 .
Let's factor the quadratic expression:
x 2 + x − 6 = ( x + 3 ) ( x − 2 )
So, the denominator is zero when x = − 3 or x = 2 . Since we have the condition x < 3 , both x = − 3 and x = 2 are within this interval and must be excluded from the domain.
Determine the domain of the first part Therefore, the domain of the first part of the function is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) .
Analyze the second part of the function For the second part of the function, g ( x ) = lo g 2 ( x + 5 ) for x ≥ 3 , we need to consider the argument of the logarithm. The argument must be greater than zero, so we need to find the values of x for which 0"> x + 5 > 0 .
Solving this inequality, we get:
-5"> x > − 5
Since we also have the condition x ≥ 3 , the domain of the second part of the function is [ 3 , ∞ ) .
Combine the domains Now, we need to combine the domains of both parts of the function. The domain of the first part is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) , and the domain of the second part is [ 3 , ∞ ) .
Combining these, we get the domain of the entire piecewise function as ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) ∪ [ 3 , ∞ ) . This can be simplified to ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .
Final Answer Therefore, the domain of the piecewise function g ( x ) is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .
Examples
Piecewise functions are useful in modeling situations where different rules apply over different intervals. For example, consider a cell phone plan where you pay a fixed monthly fee for a certain amount of data, and then you pay an additional fee for each gigabyte of data you use beyond that limit. The cost function can be expressed as a piecewise function. Another example is calculating income tax, where different tax rates apply to different income brackets. Understanding the domain of such functions helps in determining the range of inputs for which the function is valid and provides meaningful outputs.
The domain of the piecewise function is determined by evaluating both parts. The first part is defined for ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , 3 ) , and the second part is defined for [ 3 , ∞ ) . Thus, the complete domain is ( − ∞ , − 3 ) ∪ ( − 3 , 2 ) ∪ ( 2 , ∞ ) .
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