If [tex]f(x)=2 x^2+5 \sqrt{(x-2)}[/tex], complete the following statement:
[tex]f(6)=[/tex]
Answer (1)
Substitute x = 6 into the function f ( x ) = 2 x 2 + 5 ( x − 2 ) .
Calculate f ( 6 ) = 2 ( 6 ) 2 + 5 ( 6 − 2 ) .
Simplify the expression: f ( 6 ) = 2 ( 36 ) + 5 4 = 72 + 5 ( 2 ) = 72 + 10 .
Obtain the final result: f ( 6 ) = 82 , so the answer is 82 .
Explanation
Understanding the problem We are given the function f ( x ) = 2 x 2 + 5 ( x − 2 ) and we want to find the value of f ( 6 ) . This means we need to substitute x = 6 into the function and simplify.
Substituting x=6 Now, let's substitute x = 6 into the function: f ( 6 ) = 2 ( 6 ) 2 + 5 ( 6 − 2 )
Simplifying the expression Next, we simplify the expression. First, we calculate 6 2 which is 36 . Then we multiply by 2: 2 × 36 = 72 . So we have: f ( 6 ) = 72 + 5 ( 6 − 2 )
Simplifying the square root Now, we simplify the square root. We have 6 − 2 = 4 , so ( 6 − 2 ) = 4 = 2 . Then we multiply by 5: 5 × 2 = 10 . So we have: f ( 6 ) = 72 + 10
Final Calculation Finally, we add the two numbers: 72 + 10 = 82 . Therefore, f ( 6 ) = 82 .
Final Answer Therefore, f ( 6 ) = 82 .
Examples
Understanding functions is crucial in many real-world applications. For example, if you are tracking the growth of a plant, you might use a function to model its height over time. If f ( x ) = 2 x 2 + 5 x − 2 represents the height of a plant (in cm) x days after planting (where x ≥ 2 ), then finding f ( 6 ) tells you the height of the plant 6 days after planting. In this case, the plant would be 82 cm tall. This kind of modeling helps in agriculture, environmental science, and even in predicting population growth.
