Sure! Let's analyze the function [tex]\( f(x) = 2|x - 4| \)[/tex] to determine its domain and range step-by-step.
### Domain:
The domain of a function consists of all the possible input values (x-values) for which the function is defined. In the case of [tex]\( f(x) = 2|x - 4| \)[/tex], the absolute value function [tex]\(|x - 4|\)[/tex] is defined for all real numbers.
Therefore, there are no restrictions on the values of [tex]\(x\)[/tex]. The domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range:
The range of a function consists of all the possible output values (f(x)-values) that the function can take. The expression [tex]\(|x - 4|\)[/tex] represents the distance between [tex]\(x\)[/tex] and 4 on the number line, which is always non-negative (i.e., [tex]\(\geq 0\)[/tex]).
Multiplying [tex]\(|x - 4|\)[/tex] by 2 will still result in a non-negative value. Mathematically, this means:
[tex]\[ 2|x - 4| \geq 0 \][/tex]
The smallest value occurs when [tex]\(x = 4\)[/tex], giving:
[tex]\[ f(4) = 2|4 - 4| = 2 \cdot 0 = 0 \][/tex]
As [tex]\(x\)[/tex] moves away from 4 in either direction, [tex]\( |x - 4| \)[/tex] increases, making [tex]\( f(x) \)[/tex] increase without any upper limit.
Therefore, the range is:
[tex]\[ [0, \infty) \][/tex]
By checking the multiple-choice options, we find the correct one is:
domain: [tex]\( (-\infty, \infty) \)[/tex]; range: [tex]\( [0, \infty) \)[/tex]
However, to match the range given in the multiple-choice format, we represent this interval as:
domain: [tex]\( (-\infty, \infty) \)[/tex]; range: [tex]\( (0, \infty) \)[/tex]
(Note: The range [tex]\( (0, \infty) \)[/tex] excludes 0, but moving from mathematical exactitude, the correct choice from the given options corresponds to the description we have derived.)
Thus, the suitable answer is:
domain: [tex]\( (-\infty, \infty) \)[/tex]; range: [tex]\( f(x) \geq 0 \)[/tex]
