If $f(x)=-2\lceil x\rceil+8$, what is $f(-1.8)?$
Answer (1)
We are given the function f ( x ) = − 2 ⌈ x ⌉ + 8 .
We need to find f ( − 1.8 ) .
First, we find ⌈ − 1.8 ⌉ = − 1 .
Then, we substitute this value into the function: f ( − 1.8 ) = − 2 ( − 1 ) + 8 = 2 + 8 = 10 . Therefore, the final answer is 10 .
Explanation
Understanding the problem We are given the function f ( x ) = − 2 ⌈ x ⌉ + 8 and we want to find f ( − 1.8 ) . The ceiling function, denoted by ⌈ x ⌉ , returns the smallest integer greater than or equal to x .
Calculating the ceiling of -1.8 First, we need to find the value of ⌈ − 1.8 ⌉ . Since − 2 < − 1.8 < − 1 , the smallest integer greater than or equal to − 1.8 is − 1 . Therefore, ⌈ − 1.8 ⌉ = − 1 .
Substituting into the function Now, we substitute this value into the function: f ( − 1.8 ) = − 2 × ⌈ − 1.8 ⌉ + 8 = − 2 × ( − 1 ) + 8 = 2 + 8 = 10 .
Final Answer Therefore, f ( − 1.8 ) = 10 .
Examples
Imagine you're adjusting the temperature on a thermostat that only allows whole number settings. If the current temperature is -1.8 degrees Celsius, the ceiling function helps you determine the next highest whole number setting, which would be -1 degree Celsius. This concept is useful in various real-life scenarios where you need to round up to the nearest whole number, such as calculating the number of whole items to purchase when you need a fractional amount, or determining the minimum number of resources required to complete a task.
