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The function [tex][tex]$g(x) = |x - 6| - 8$[/tex][/tex] is graphed. What is the range?

A. [tex]\{ y \mid y \ \textgreater \ -8 \}[/tex]
B. [tex]\{ y \mid y \geq -8 \}[/tex]
C. [tex]\{ y \mid y \ \textless \ -8 \}[/tex]
D. [tex]\{ y \mid y \text{ is all real numbers} \}[/tex]



Answer :

GinnyAnswer
To determine the range of the function [tex]\( g(x) = |x-6| - 8 \)[/tex], we need to understand the behavior and properties of the absolute value function.

1. Absolute Value Function: The expression [tex]\( |x-6| \)[/tex] represents the absolute value of [tex]\( x-6 \)[/tex], which measures the distance between [tex]\( x \)[/tex] and 6 on the number line. Therefore, [tex]\( |x-6| \)[/tex] is always greater than or equal to 0 for all real numbers [tex]\( x \)[/tex]:
[tex]\[ |x-6| \geq 0 \quad \text{for all} \quad x \][/tex]

2. Subtraction Outside the Absolute Value: Next, we adjust this by subtracting 8:
[tex]\[ g(x) = |x-6| - 8 \][/tex]
Since [tex]\( |x-6| \geq 0 \)[/tex], when we subtract 8, we need to adjust our understanding of the new minimum value of [tex]\( g(x) \)[/tex]. The smallest value [tex]\( |x-6| \)[/tex] can take is 0 (when [tex]\( x = 6 \)[/tex]), leading to:
[tex]\[ g(x) = |6-6| - 8 = 0 - 8 = -8 \][/tex]
Therefore, the smallest value [tex]\( g(x) \)[/tex] can be is [tex]\( -8 \)[/tex].

3. Other Values: As [tex]\( |x-6| \)[/tex] increases beyond 0, [tex]\( g(x) \)[/tex] will increase correspondingly. For example:
[tex]\[ \text{If } x = 5, \quad g(x) = |5-6| - 8 = 1 - 8 = -7 \][/tex]
[tex]\[ \text{If } x = 8, \quad g(x) = |8-6| - 8 = 2 - 8 = -6 \][/tex]
This shows that as we take values of [tex]\( x \)[/tex] farther from 6, [tex]\( g(x) \)[/tex] increases without bound.

4. Range of [tex]\( g(x) \)[/tex]: Combining these observations, the function [tex]\( g(x) = |x-6| - 8 \)[/tex] takes on all values from [tex]\( -8 \)[/tex] upwards. In mathematical notation, this means:
[tex]\[ y \geq -8 \][/tex]

Therefore, the correct answer is:
[tex]\[ \{ y \mid y \geq -8 \} \][/tex]
Intriguing456

Answer:

C) [tex]\text{range} = \{y\, \left|\frac{}{}y\ge-8\}[/tex]

Step-by-step explanation:

We are finding the range, or set of outputs, for the absolute value function:

[tex]g(x) = |x - 6| - 8[/tex]

We can rewrite this function using piecewise notation:

[tex]g(x) = \begin{cases}(x-6) - 8 & \text{if } x \geq 6 \\-(x-6) - 8 & \text{if } x < 6\end{cases}[/tex]

↓↓↓

[tex]g(x) = \begin{cases}x - 14 & \text{if } x \geq 6 \\- x - 2 & \text{if } x < 6\end{cases}[/tex]

Since both sides are symmetrical about x = 6, we can examine one piece and assume the range is the same for the other piece.

We know that the upper piece is an increasing function. Therefore, the lowest that the upper piece will ever be is at the leftmost point, which is the vertex of the function. The vertex happens when:

  • x = 6

at which point, the y-value is:

  • y = 6 - 14 = -8

Again, since the upper piece is always increasing, it will go until infinity. Therefore, the range of the function is:

C) [tex]\text{range} = \{y\, \left|\frac{}{}y\ge-8\}[/tex]

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