Answer :
To determine if the two functions [tex]\( F(x) = \sqrt{x} - 6 \)[/tex] and [tex]\( G(x) = (x + 6)^2 \)[/tex] are inverses of each other, we need to check the compositions [tex]\( F(G(x)) \)[/tex] and [tex]\( G(F(x)) \)[/tex]. If [tex]\( F(G(x)) = x \)[/tex] and [tex]\( G(F(x)) = x \)[/tex], then the functions are inverses of each other.
### Step 1: Compute [tex]\( F(G(x)) \)[/tex]:
1. Substitute [tex]\( G(x) \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F(G(x)) = F((x + 6)^2) \][/tex]
2. Evaluate [tex]\( F((x + 6)^2) \)[/tex]:
[tex]\[ F((x + 6)^2) = \sqrt{(x + 6)^2} - 6 \][/tex]
Recall that [tex]\( \sqrt{(x + 6)^2} \)[/tex] simplifies to [tex]\( |x + 6| \)[/tex]. So,
[tex]\[ F((x + 6)^2) = |x + 6| - 6 \][/tex]
### Step 2: Compute [tex]\( G(F(x)) \)[/tex]:
1. Substitute [tex]\( F(x) \)[/tex] into [tex]\( G(x) \)[/tex]:
[tex]\[ G(F(x)) = G(\sqrt{x} - 6) \][/tex]
2. Evaluate [tex]\( G(\sqrt{x} - 6) \)[/tex]:
[tex]\[ G(\sqrt{x} - 6) = (\sqrt{x} - 6 + 6)^2 \][/tex]
Simplifying inside the parenthesis:
[tex]\[ G(\sqrt{x} - 6) = \sqrt{x}^2 \][/tex]
Which simplifies to:
[tex]\[ G(\sqrt{x} - 6) = x \][/tex]
### Step 3: Check the Simplified Results:
- For [tex]\( F(G(x)) \)[/tex]:
[tex]\[ F(G(x)) = |x + 6| - 6 \][/tex]
Notice that [tex]\( |x + 6| \)[/tex] depends on whether [tex]\( x + 6 \)[/tex] is positive or negative, but in either case, [tex]\( |x + 6| \neq x + 6 \)[/tex] for all [tex]\( x \)[/tex]. Thus:
[tex]\[ F(G(x)) \neq x \][/tex]
- For [tex]\( G(F(x)) \)[/tex]:
[tex]\[ G(F(x)) = x \][/tex]
### Conclusion:
Since [tex]\( F(G(x)) \neq x \)[/tex] even though [tex]\( G(F(x)) = x \)[/tex], the functions [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] are not inverses of each other.
The detailed compositions are:
- [tex]\( F(G(x)) = |x+6| - 6 \)[/tex]
- [tex]\( G(F(x)) = x \)[/tex]
Therefore, [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] are not inverses of each other.
### Step 1: Compute [tex]\( F(G(x)) \)[/tex]:
1. Substitute [tex]\( G(x) \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F(G(x)) = F((x + 6)^2) \][/tex]
2. Evaluate [tex]\( F((x + 6)^2) \)[/tex]:
[tex]\[ F((x + 6)^2) = \sqrt{(x + 6)^2} - 6 \][/tex]
Recall that [tex]\( \sqrt{(x + 6)^2} \)[/tex] simplifies to [tex]\( |x + 6| \)[/tex]. So,
[tex]\[ F((x + 6)^2) = |x + 6| - 6 \][/tex]
### Step 2: Compute [tex]\( G(F(x)) \)[/tex]:
1. Substitute [tex]\( F(x) \)[/tex] into [tex]\( G(x) \)[/tex]:
[tex]\[ G(F(x)) = G(\sqrt{x} - 6) \][/tex]
2. Evaluate [tex]\( G(\sqrt{x} - 6) \)[/tex]:
[tex]\[ G(\sqrt{x} - 6) = (\sqrt{x} - 6 + 6)^2 \][/tex]
Simplifying inside the parenthesis:
[tex]\[ G(\sqrt{x} - 6) = \sqrt{x}^2 \][/tex]
Which simplifies to:
[tex]\[ G(\sqrt{x} - 6) = x \][/tex]
### Step 3: Check the Simplified Results:
- For [tex]\( F(G(x)) \)[/tex]:
[tex]\[ F(G(x)) = |x + 6| - 6 \][/tex]
Notice that [tex]\( |x + 6| \)[/tex] depends on whether [tex]\( x + 6 \)[/tex] is positive or negative, but in either case, [tex]\( |x + 6| \neq x + 6 \)[/tex] for all [tex]\( x \)[/tex]. Thus:
[tex]\[ F(G(x)) \neq x \][/tex]
- For [tex]\( G(F(x)) \)[/tex]:
[tex]\[ G(F(x)) = x \][/tex]
### Conclusion:
Since [tex]\( F(G(x)) \neq x \)[/tex] even though [tex]\( G(F(x)) = x \)[/tex], the functions [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] are not inverses of each other.
The detailed compositions are:
- [tex]\( F(G(x)) = |x+6| - 6 \)[/tex]
- [tex]\( G(F(x)) = x \)[/tex]
Therefore, [tex]\( F(x) \)[/tex] and [tex]\( G(x) \)[/tex] are not inverses of each other.
