If $f(x)=|x|+9$ and $g(x)=-6$, which describes the range of $(f+g)(x)$?
A. $(f+g)(x) \geq 3$ for all values of $x$
B. $(f+g)(x) \leq 3$ for all values of $x$
C. $(f+g)(x) \leq 6$ for all values of $x$
D. $(f+g)(x) \geq 6$ for all values of $x$
Answer (1)
Find the combined function: ( f + g ) ( x ) = ∣ x ∣ + 3 .
Analyze the absolute value: ∣ x ∣ ≥ 0 .
Determine the range: ( f + g ) ( x ) ≥ 3 .
The range of ( f + g ) ( x ) is ( f + g ) ( x ) ≥ 3 for all x , so the answer is ( f + g ) ( x ) ≥ 3 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = ∣ x ∣ + 9 and g ( x ) = − 6 . We want to find the range of the function ( f + g ) ( x ) . The range of a function is the set of all possible output values.
Finding the Combined Function First, we need to find the expression for ( f + g ) ( x ) . This is done by adding the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = ∣ x ∣ + 9 + ( − 6 ) = ∣ x ∣ + 3 So, ( f + g ) ( x ) = ∣ x ∣ + 3 .
Analyzing the Absolute Value Function Next, we analyze the range of the absolute value function ∣ x ∣ . The absolute value of any real number is always non-negative, meaning =" 0"> ∣ x ∣" >= "0 for all x .
Determining the Range of (f+g)(x) Now, we determine the range of ( f + g ) ( x ) = ∣ x ∣ + 3 . Since =" 0"> ∣ x ∣" >= "0 , we can add 3 to both sides of the inequality: =" 0 + 3"> ∣ x ∣ + 3" >= "0 + 3 =" 3"> ( f + g ) ( x ) " >= "3 This means that the smallest possible value of ( f + g ) ( x ) is 3, and it can take any value greater than or equal to 3.
Conclusion Finally, we compare our result with the given options. We found that =" 3"> ( f + g ) ( x ) " >= "3 for all values of x . This matches the first option.
Examples
Imagine you're tracking the temperature changes in a freezer. The function f ( x ) = ∣ x ∣ + 9 represents the temperature fluctuations, where ∣ x ∣ is the absolute change from a set point, and 9 is the base temperature. If the freezer has a constant cooling effect represented by g ( x ) = − 6 , then ( f + g ) ( x ) shows the actual temperature range. Understanding this range ensures the freezer stays above a critical threshold, like 3 degrees, to prevent spoilage. This helps in practical applications like food storage, where maintaining a minimum temperature is crucial.
