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

Combine the functions: ( f + g ) ( x ) = ∣ x ∣ + 9 − 6 = ∣ x ∣ + 3 .
Determine the minimum value of ∣ x ∣ , which is 0.
Find the minimum value of ( f + g ) ( x ) : ∣ x ∣ + 3 ≥ 3 .
The range of ( f + g ) ( x ) 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.

Combining the Functions 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 )

Simplifying the Expression Now, we simplify the expression: ( f + g ) ( x ) = ∣ x ∣ + 3

Finding the Minimum Value of |x| Next, we need to determine the minimum value of ∣ x ∣ . The absolute value of any real number is non-negative, meaning ∣ x ∣ ≥ 0 for all x . The minimum value of ∣ x ∣ is 0, which occurs when x = 0 .

Finding the Minimum Value of (f+g)(x) Now, we determine the minimum value of ( f + g ) ( x ) . Since the minimum value of ∣ x ∣ is 0, the minimum value of ( f + g ) ( x ) = ∣ x ∣ + 3 is 0 + 3 = 3 . Since ∣ x ∣ can take any non-negative value, ( f + g ) ( x ) can take any value greater than or equal to 3.

Determining the Range Therefore, the range of ( f + g ) ( x ) is all values greater than or equal to 3. We can express this as an inequality: ( f + g ) ( x ) ≥ 3 .


Examples
Understanding function ranges is crucial in many real-world applications. For example, if f ( x ) represents the cost of producing x items and g ( x ) represents a fixed discount, ( f + g ) ( x ) would represent the final cost. Knowing the range of ( f + g ) ( x ) helps determine the minimum cost you'll ever have to pay, which is essential for budgeting and financial planning. This concept applies in various fields, from economics to engineering, where understanding the bounds of a function is vital for making informed decisions.

Answered by GinnyAnswer | 2025-07-03

The range of ( f + g ) ( x ) is all values greater than or equal to 3, which means that ( f + g ) ( x ) ≥ 3 . Hence, the correct answer is option A.
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Answered by Anonymous | 2025-07-04