Answer :
To determine if two functions are inverses of each other, we need to check if their compositions [tex]\( F(G(x)) \)[/tex] and [tex]\( G(F(x)) \)[/tex] both simplify to [tex]\( x \)[/tex].
Given the functions:
[tex]\[ F(x) = \sqrt{x} + 3 \][/tex]
[tex]\[ G(x) = (x - 3)^2 \][/tex]
Step-by-step solution:
1. Compute the composition [tex]\( F(G(x)) \)[/tex]:
[tex]\( G(x) = (x - 3)^2 \)[/tex]
Now substitute [tex]\( G(x) \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F(G(x)) = F((x - 3)^2) = \sqrt{(x - 3)^2} + 3 \][/tex]
Since [tex]\(\sqrt{(x - 3)^2} \)[/tex] is [tex]\(|x - 3|\)[/tex]:
[tex]\[ F(G(x)) = |x - 3| + 3 \][/tex]
For [tex]\( x \geq 3 \)[/tex], [tex]\(|x - 3| = x - 3\)[/tex], so:
[tex]\[ F(G(x)) = x - 3 + 3 = x \][/tex]
For [tex]\( x < 3 \)[/tex], [tex]\(|x - 3| = -(x - 3) = 3 - x\)[/tex], so:
[tex]\[ F(G(x)) = 3 - x + 3 = 6 - x \][/tex]
Therefore:
[tex]\[ F(G(x)) = |x - 3| + 3 \][/tex]
2. Compute the composition [tex]\( G(F(x)) \)[/tex]:
[tex]\( F(x) = \sqrt{x} + 3 \)[/tex]
Now substitute [tex]\( F(x) \)[/tex] into [tex]\( G(x) \)[/tex]:
[tex]\[ G(F(x)) = G(\sqrt{x} + 3) = (\sqrt{x} + 3 - 3)^2 \][/tex]
Simplifying inside the parentheses:
[tex]\[ G(F(x)) = (\sqrt{x})^2 \][/tex]
Which simplifies to:
[tex]\[ G(F(x)) = x \][/tex]
From the compositions, we observe the following:
[tex]\[ F(G(x)) = |x - 3| + 3 \quad \text{and} \quad G(F(x)) = x \][/tex]
Since [tex]\( F(G(x)) \neq x \)[/tex] for all [tex]\( x \)[/tex] (specifically because of the absolute value operation), the composition [tex]\( F(G(x)) \)[/tex] does not result in [tex]\( x \)[/tex] consistently.
Therefore, the correct answer is:
[tex]\[ \text{A. No, because the composition does not result in an answer of } x. \][/tex]
Given the functions:
[tex]\[ F(x) = \sqrt{x} + 3 \][/tex]
[tex]\[ G(x) = (x - 3)^2 \][/tex]
Step-by-step solution:
1. Compute the composition [tex]\( F(G(x)) \)[/tex]:
[tex]\( G(x) = (x - 3)^2 \)[/tex]
Now substitute [tex]\( G(x) \)[/tex] into [tex]\( F(x) \)[/tex]:
[tex]\[ F(G(x)) = F((x - 3)^2) = \sqrt{(x - 3)^2} + 3 \][/tex]
Since [tex]\(\sqrt{(x - 3)^2} \)[/tex] is [tex]\(|x - 3|\)[/tex]:
[tex]\[ F(G(x)) = |x - 3| + 3 \][/tex]
For [tex]\( x \geq 3 \)[/tex], [tex]\(|x - 3| = x - 3\)[/tex], so:
[tex]\[ F(G(x)) = x - 3 + 3 = x \][/tex]
For [tex]\( x < 3 \)[/tex], [tex]\(|x - 3| = -(x - 3) = 3 - x\)[/tex], so:
[tex]\[ F(G(x)) = 3 - x + 3 = 6 - x \][/tex]
Therefore:
[tex]\[ F(G(x)) = |x - 3| + 3 \][/tex]
2. Compute the composition [tex]\( G(F(x)) \)[/tex]:
[tex]\( F(x) = \sqrt{x} + 3 \)[/tex]
Now substitute [tex]\( F(x) \)[/tex] into [tex]\( G(x) \)[/tex]:
[tex]\[ G(F(x)) = G(\sqrt{x} + 3) = (\sqrt{x} + 3 - 3)^2 \][/tex]
Simplifying inside the parentheses:
[tex]\[ G(F(x)) = (\sqrt{x})^2 \][/tex]
Which simplifies to:
[tex]\[ G(F(x)) = x \][/tex]
From the compositions, we observe the following:
[tex]\[ F(G(x)) = |x - 3| + 3 \quad \text{and} \quad G(F(x)) = x \][/tex]
Since [tex]\( F(G(x)) \neq x \)[/tex] for all [tex]\( x \)[/tex] (specifically because of the absolute value operation), the composition [tex]\( F(G(x)) \)[/tex] does not result in [tex]\( x \)[/tex] consistently.
Therefore, the correct answer is:
[tex]\[ \text{A. No, because the composition does not result in an answer of } x. \][/tex]
