Which function has the same range as [tex]f(x)=-2 \sqrt{x-3}+8[/tex]?
A. [tex]g(x)=\sqrt{x-3}-8[/tex]
B. [tex]g(x)=\sqrt{x-3}+8[/tex]
C. [tex]g(x)=-\sqrt{x+3}+8[/tex]
D. [tex]g(x)=-\sqrt{x-3}-8[/tex]
Answer (2)
Determine the range of f ( x ) = − 2 x − 3 + 8 , which is ( − ∞ , 8 ] .
Find the range of each g ( x ) function.
Compare the ranges of the g ( x ) functions to the range of f ( x ) .
The function g ( x ) = − x + 3 + 8 has the same range as f ( x ) . The answer is g ( x ) = − x + 3 + 8 .
Explanation
Understanding the Problem We are given the function f ( x ) = − 2 x − 3 + 8 and asked to find which of the given functions g ( x ) has the same range as f ( x ) . The range of a function is the set of all possible output values (y-values) that the function can produce.
Finding the Range of f(x) First, let's determine the range of f ( x ) . The square root function x − 3 is defined for x ≥ 3 , and its output is always non-negative, i.e., x − 3 ≥ 0 . Multiplying by -2, we get − 2 x − 3 ≤ 0 . Adding 8, we have − 2 x − 3 + 8 ≤ 8 . Therefore, the range of f ( x ) is all real numbers less than or equal to 8, which can be written as ( − ∞ , 8 ] .
Finding the Ranges of g(x) Functions Now, let's find the range of each of the given g ( x ) functions:
g ( x ) = x − 3 − 8 . Since x − 3 ≥ 0 for x ≥ 3 , we have x − 3 − 8 ≥ − 8 . The range of this g ( x ) is [ − 8 , ∞ ) .
g ( x ) = x − 3 + 8 . Since x − 3 ≥ 0 for x ≥ 3 , we have x − 3 + 8 ≥ 8 . The range of this g ( x ) is [ 8 , ∞ ) .
g ( x ) = − x + 3 + 8 . Since x + 3 ≥ 0 for x ≥ − 3 , we have − x + 3 ≤ 0 . Thus, − x + 3 + 8 ≤ 8 . The range of this g ( x ) is ( − ∞ , 8 ] .
g ( x ) = − x − 3 − 8 . Since x − 3 ≥ 0 for x ≥ 3 , we have − x − 3 ≤ 0 . Thus, − x − 3 − 8 ≤ − 8 . The range of this g ( x ) is ( − ∞ , − 8 ] .
Comparing the Ranges Comparing the ranges, we see that the range of f ( x ) is ( − ∞ , 8 ] , and the range of g ( x ) = − x + 3 + 8 is also ( − ∞ , 8 ] . Therefore, the function g ( x ) = − x + 3 + 8 has the same range as f ( x ) .
Final Answer The function with the same range as f ( x ) = − 2 x − 3 + 8 is g ( x ) = − x + 3 + 8 .
Examples
Understanding the range of functions is crucial in many real-world applications. For example, if you're designing a suspension bridge, you need to know the range of possible stresses and strains on the materials to ensure the bridge's safety. Similarly, in economics, understanding the range of possible outcomes for an investment can help you make informed decisions. In physics, when analyzing projectile motion, knowing the range of the projectile helps determine where it will land. These examples show how understanding function ranges is essential for solving practical problems in various fields.
The function with the same range as f ( x ) = − 2 x − 3 + 8 is g ( x ) = − x + 3 + 8 because both have the range ( − ∞ , 8 ] .
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