Question

If f(x) = |x| + 9 and g(x) = –6, which of the following statements accurately describes the value of (f + g)(x)?

A) (f + g)(x) is greater than or equal to 3 for all values of x.
B) (f + g)(x) is less than or equal to 3 for all values of x.
C) (f + g)(x) is less than or equal to 6 for all values of x.
D) (f + g)(x) is greater than or equal to 6 for all values of x.

Answer 1

The sum of the functions f(x) = |x| + 9 and g(x) = -6 is (f + g)(x) = |x| + 3. Since the absolute value of x is always non-negative, (f + g)(x) is always greater than or equal to 3 for all x. Thus, the correct answer is A) (f + g)(x) is greater than or equal to 3 for all values of x.

Step-by-step explanation:

To determine the value of (f + g)(x), we need to add the functions f(x) and g(x) together. Given that f(x) = |x| + 9 and g(x) = –6, we can find the sum of these two functions:

(f + g)(x) = f(x) + g(x) = (|x| + 9) + (-6) = |x| + 3

Since the absolute value function |x| is always greater than or equal to 0, the smallest value (f + g)(x) can take is when x = 0, which would make |x| = 0, and thus:

(f + g)(0) = |0| + 3 = 0 + 3 = 3

Therefore, for all values of x, the sum (f + g)(x) will be greater than or equal to 3. This means the correct choice is:

A) (f + g)(x) is greater than or equal to 3 for all values of x.