The same function is used, but the \(a\) value is increased by 2. How do the domain and range of the new function compare to the domain and range of the original function?
A. The range stays the same.
B. The range becomes \(y>2\).
C. The domain stays the same.
D. The domain becomes \(x>2\).
E. The range becomes \(y \geq 2\).
F. The domain becomes \(x>\) ?
Answer (1)
Increasing the 'a' value in a function can affect its domain and range.
The effect depends on the specific function.
For a quadratic function f ( x ) = x 2 + a , increasing 'a' by 2 shifts the range upwards by 2.
If the original range was y ≥ 0 , the new range becomes y ≥ 2 , while the domain remains unchanged. y ≥ 2 .
Explanation
Problem Analysis Let's analyze the problem. We are given that the 'a' value of a function is increased by 2. We need to determine how this change affects the domain and range of the function. The question provides several possible answers regarding the changes to the domain and range. We need to identify the correct statement.
Function Examples To understand the impact of changing the 'a' value, let's consider a few examples of functions and how their domain and range are affected.
Linear Function: If f ( x ) = a x + b , increasing 'a' by 2 gives g ( x ) = ( a + 2 ) x + b . The domain and range of both functions are all real numbers, so they remain unchanged.
Square Root Function: If f ( x ) = a x , increasing 'a' by 2 gives g ( x ) = ( a + 2 ) x . The domain is x ≥ 0 and the range is y ≥ 0 for both functions, so they remain unchanged.
Rational Function: If f ( x ) = x − a 1 , increasing 'a' by 2 gives g ( x ) = x − ( a + 2 ) 1 . The domain of f ( x ) is x = a , and the range is y = 0 . The domain of g ( x ) is x = a + 2 , and the range is y = 0 . The range stays the same, but the domain changes.
Quadratic Function: If f ( x ) = x 2 + a , increasing 'a' by 2 gives g ( x ) = x 2 + a + 2 . The domain is all real numbers. The range of f ( x ) is y ≥ a , and the range of g ( x ) is y ≥ a + 2 . The domain stays the same, but the range changes.
Analyzing the Options Based on the examples, we can see that the effect on the domain and range depends on the specific function. Let's examine the given options:
The range stays the same.
The range becomes 2"> y > 2 .
The domain stays the same.
The domain becomes 2"> x > 2 .
The range becomes y ≥ 2 .
The domain becomes "> x > ?
From the quadratic function example, we saw that the domain can stay the same while the range changes. Specifically, if f ( x ) = x 2 + a , then g ( x ) = x 2 + a + 2 . The range of f ( x ) is y ≥ a , and the range of g ( x ) is y ≥ a + 2 . If 'a' was 0, the range would become y ≥ 2 .
Determining the Correct Statement Considering the quadratic function example f ( x ) = x 2 + a , when 'a' is increased by 2, the range becomes y ≥ a + 2 . If the original range was y ≥ a , the new range is shifted upwards by 2. If we assume the original function had a range of y ≥ 0 (i.e., a = 0 ), then the new range would be y ≥ 2 . The domain remains all real numbers.
Therefore, the correct statement is that the range becomes y ≥ 2 and the domain stays the same.
Examples
Consider a simple electronic circuit where the output voltage is determined by a function of the input voltage, V o u t = V in 2 + a . If we increase the value of 'a' by 2, the new output voltage becomes V o u t = V in 2 + a + 2 . This means the entire output voltage range is shifted upwards by 2 volts. Understanding how changing 'a' affects the range is crucial in designing and analyzing electronic circuits to ensure the output voltage stays within desired limits.
