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Answer :
We begin by noting that the logarithm is only defined for positive arguments. Thus, for the function
[tex]$$
f(x)=\log_{10}(6-|1-x|)
$$[/tex]
the expression inside the logarithm must be positive:
[tex]$$
6-|1-x| > 0.
$$[/tex]
This inequality can be rearranged as:
[tex]$$
|1-x| < 6.
$$[/tex]
The inequality involving the absolute value can be written as a double inequality:
[tex]$$
-6 < 1-x < 6.
$$[/tex]
Now, we solve each part of the inequality separately.
1. For the left part:
[tex]$$
-6 < 1-x
$$[/tex]
Subtract 1 from both sides:
[tex]$$
-6-1 < -x \quad \Longrightarrow \quad -7 < -x.
$$[/tex]
Multiplying both sides by [tex]$-1$[/tex] (which reverses the inequality):
[tex]$$
7 > x \quad \Longrightarrow \quad x < 7.
$$[/tex]
2. For the right part:
[tex]$$
1-x < 6
$$[/tex]
Subtract 1 from both sides:
[tex]$$
-x < 5.
$$[/tex]
Again, multiply both sides by [tex]$-1$[/tex], remembering to reverse the inequality:
[tex]$$
x > -5.
$$[/tex]
Thus, we obtain the combined inequality:
[tex]$$
-5 < x < 7.
$$[/tex]
Therefore, the domain of the function is all real numbers [tex]$x$[/tex] such that:
[tex]$$
-5 < x < 7.
$$[/tex]
Among the provided options, the interval [tex]$-5
[tex]$$
f(x)=\log_{10}(6-|1-x|)
$$[/tex]
the expression inside the logarithm must be positive:
[tex]$$
6-|1-x| > 0.
$$[/tex]
This inequality can be rearranged as:
[tex]$$
|1-x| < 6.
$$[/tex]
The inequality involving the absolute value can be written as a double inequality:
[tex]$$
-6 < 1-x < 6.
$$[/tex]
Now, we solve each part of the inequality separately.
1. For the left part:
[tex]$$
-6 < 1-x
$$[/tex]
Subtract 1 from both sides:
[tex]$$
-6-1 < -x \quad \Longrightarrow \quad -7 < -x.
$$[/tex]
Multiplying both sides by [tex]$-1$[/tex] (which reverses the inequality):
[tex]$$
7 > x \quad \Longrightarrow \quad x < 7.
$$[/tex]
2. For the right part:
[tex]$$
1-x < 6
$$[/tex]
Subtract 1 from both sides:
[tex]$$
-x < 5.
$$[/tex]
Again, multiply both sides by [tex]$-1$[/tex], remembering to reverse the inequality:
[tex]$$
x > -5.
$$[/tex]
Thus, we obtain the combined inequality:
[tex]$$
-5 < x < 7.
$$[/tex]
Therefore, the domain of the function is all real numbers [tex]$x$[/tex] such that:
[tex]$$
-5 < x < 7.
$$[/tex]
Among the provided options, the interval [tex]$-5
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