To determine which pair of functions are inverses of one another, we need to verify if composing one function with the other returns the original input.
### Step-by-Step Solution:
Let's evaluate each pair of functions to tell if they are inverses.
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#### Pair 1:
1. [tex]\( y = \log_2(x) - \log_2(12) \)[/tex]
2. [tex]\( y = 2^{x - 12} \)[/tex]
To check if these are inverses, we'll compute [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] and see if either equals [tex]\( x \)[/tex].
##### Check [tex]\( f(g(x)) \)[/tex]:
[tex]\[ g(x) = 2^{x-12} \][/tex]
[tex]\[ f(g(x)) = \log_2(2^{x-12}) - \log_2(12) \][/tex]
[tex]\[ f(g(x)) = (x-12) - \log_2(12) \][/tex]
This does not simplify to [tex]\( x \)[/tex].
##### Check [tex]\( g(f(x)) \)[/tex]:
[tex]\[ f(x) = \log_2(x) - \log_2(12) \][/tex]
[tex]\[ g(f(x)) = 2^{(\log_2(x) - \log_2(12)) - 12} \][/tex]
This does not simplify to [tex]\( x \)[/tex].
So, Pair 1 does not consist of inverse functions.
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#### Pair 2:
1. [tex]\( y = \log_9(x) + \log_9(12) \)[/tex]
2. [tex]\( y = \frac{9^7}{12} \)[/tex]
##### Check [tex]\( f(g(x)) \)[/tex]:
[tex]\[ g(x) = \frac{9^7}{12} \][/tex]
[tex]\[ f(g(x)) = \log_9\left(\frac{9^7}{12}\right) + \log_9(12) \][/tex]
This expression does not simplify to [tex]\( x \)[/tex].
##### Check [tex]\( g(f(x)) \)[/tex]:
[tex]\[ f(x) = \log_9(x) + \log_9(12) \][/tex]
Using properties of logarithms:
[tex]\[ f(x) = \log_9(12x) \][/tex]
[tex]\[ g(f(x)) = \frac{9^7}{12} \][/tex]
This does not simplify to [tex]\( x \)[/tex].
So, Pair 2 does not consist of inverse functions.
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#### Pair 3:
1. [tex]\( y = \log_7(x)^4 \)[/tex]
2. [tex]\( y = 4 \cdot 7^x \)[/tex]
##### Check [tex]\( f(g(x)) \)[/tex]:
[tex]\[ g(x) = 4 \cdot 7^x \][/tex]
[tex]\[ f(g(x)) = (\log_7(4 \cdot 7^x))^4 \][/tex]
[tex]\[ f(g(x)) = (\log_7(4) + x)^4 \][/tex]
This does not simplify to [tex]\( x \)[/tex].
##### Check [tex]\( g(f(x)) \)[/tex]:
[tex]\[ f(x) = (\log_7(x))^4 \][/tex]
[tex]\[ g(f(x)) = 4 \cdot 7^{(\log_7(x))^4} \][/tex]
This does not simplify to [tex]\( x \)[/tex].
So, Pair 3 does not consist of inverse functions.
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#### Pair 4:
1. [tex]\( y = \log_5(x) + \log_5(7) \)[/tex]
2. [tex]\( y = 5^x + 5^7 \)[/tex]
##### Check [tex]\( f(g(x)) \)[/tex]:
[tex]\[ g(x) = 5^x + 5^7 \][/tex]
[tex]\[ f(g(x)) = \log_5(5^x + 5^7) + \log_5(7) \][/tex]
This does not simplify to [tex]\( x \)[/tex].
##### Check [tex]\( g(f(x)) \)[/tex]:
[tex]\[ f(x) = \log_5(x) + \log_5(7) \][/tex]
Using properties of logarithms:
[tex]\[ f(x) = \log_5(7x) \][/tex]
[tex]\[ g(f(x)) = 5^{\log_5(7x)} + 5^7 \][/tex]
[tex]\[ g(f(x)) = 7x + 5^7 \][/tex]
This does not simplify to [tex]\( x \)[/tex].
So, Pair 4 does not consist of inverse functions.
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After careful examination, none of the given pairs of functions meet the criteria of being inverse functions of each other. Therefore, the correct conclusion is that there is no pair of functions among the given options that are inverses of each other.
Thus, the answer is [tex]\(-1\)[/tex]: No pair of inverses found.
