Answer :
To determine which pair of functions are inverses of each other, we need to check if the composition of the functions returns the identity function, which essentially means verifying two conditions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's evaluate each pair of functions:
### Pair A
Functions:
[tex]\[ f_A(x) = \frac{x}{8} + 14 \][/tex]
[tex]\[ g_A(x) = 8x - 14 \][/tex]
1. Check [tex]\( f_A(g_A(x)) \)[/tex]:
[tex]\[ f_A(g_A(x)) = f_A(8x - 14) = \frac{8x - 14}{8} + 14 = x - \frac{14}{8} + 14 = x - \frac{7}{4} + 14 \][/tex]
[tex]\[ f_A(g_A(x)) = x - 1.75 + 14 \][/tex]
[tex]\[ f_A(g_A(x)) = x + 12.25 \neq x \][/tex]
Since this does not simplify to [tex]\( x \)[/tex], they are not inverses.
2. Check [tex]\( g_A(f_A(x)) \)[/tex]:
[tex]\[ g_A(f_A(x)) = g_A\left( \frac{x}{8} + 14 \right) = 8 \left( \frac{x}{8} + 14 \right) - 14 = x + 112 - 14 = x + 98 \neq x \][/tex]
Again, this does not simplify to [tex]\( x \)[/tex], confirming they are not inverses.
### Pair B
Functions:
[tex]\[ f_B(x) = 2x^3 - 11 \][/tex]
[tex]\[ g_B(x) = \frac{x + 11}{2} \][/tex]
1. Check [tex]\( f_B(g_B(x)) \)[/tex]:
[tex]\[ f_B(g_B(x)) = f_B\left( \frac{x + 11}{2} \right) = 2\left( \frac{x + 11}{2} \right)^3 - 11 \][/tex]
This is complex and a detailed check shows that it does not simplify to [tex]\( x \)[/tex].
2. Check [tex]\( g_B(f_B(x)) \)[/tex]:
[tex]\[ g_B(f_B(x)) = g_B(2x^3 - 11) = \frac{2x^3 - 11 + 11}{2} = \frac{2x^3}{2} = x^3 \neq x \][/tex]
Thus, they are not inverses.
### Pair C
Functions:
[tex]\[ f_C(x) = 3x^3 + 15 \][/tex]
[tex]\[ g_C(x) = \sqrt[3]{\frac{x}{3}} - 15 \][/tex]
1. Check [tex]\( f_C(g_C(x)) \)[/tex]:
[tex]\[ f_C(g_C(x)) = f_C\left( \sqrt[3]{\frac{x}{3}} - 15 \right) = 3\left( \sqrt[3]{\frac{x}{3}} - 15 \right)^3 + 15 \][/tex]
Again, a detailed check is needed, and it does not simplify to [tex]\( x \)[/tex].
2. Check [tex]\( g_C(f_C(x)) \)[/tex]:
[tex]\[ g_C(f_C(x)) = g_C(3x^3 + 15) = \sqrt[3]{\frac{3x^3 + 15}{3}} - 15 = \sqrt[3]{x^3 + 5} - 15 \neq x \][/tex]
So, they are not inverses.
### Pair D
Functions:
[tex]\[ f_D(x) = \frac{3}{x} - 6 \][/tex]
[tex]\[ g_D(x) = \frac{3}{x + 6} \][/tex]
1. Check [tex]\( f_D(g_D(x)) \)[/tex]:
[tex]\[ f_D(g_D(x)) = f_D\left( \frac{3}{x + 6} \right) = \frac{3}{\frac{3}{x + 6}} - 6 = x + 6 - 6 = x \][/tex]
This simplifies to [tex]\( x \)[/tex], satisfying the condition [tex]\( f_D(g_D(x)) = x \)[/tex].
2. Check [tex]\( g_D(f_D(x)) \)[/tex]:
[tex]\[ g_D(f_D(x)) = g_D\left( \frac{3}{x} - 6 \right) = \frac{3}{\left( \frac{3}{x} \right) - 6} = \frac{3x}{3 - 6x} = \frac{3x}{3 - 6x} = x \][/tex]
This also simplifies to [tex]\( x \)[/tex], satisfying the condition [tex]\( g_D(f_D(x)) = x \)[/tex].
Thus, the functions in Pair D are indeed inverses of each other.
Answer: The pair of functions that are inverses of each other is Pair D.
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's evaluate each pair of functions:
### Pair A
Functions:
[tex]\[ f_A(x) = \frac{x}{8} + 14 \][/tex]
[tex]\[ g_A(x) = 8x - 14 \][/tex]
1. Check [tex]\( f_A(g_A(x)) \)[/tex]:
[tex]\[ f_A(g_A(x)) = f_A(8x - 14) = \frac{8x - 14}{8} + 14 = x - \frac{14}{8} + 14 = x - \frac{7}{4} + 14 \][/tex]
[tex]\[ f_A(g_A(x)) = x - 1.75 + 14 \][/tex]
[tex]\[ f_A(g_A(x)) = x + 12.25 \neq x \][/tex]
Since this does not simplify to [tex]\( x \)[/tex], they are not inverses.
2. Check [tex]\( g_A(f_A(x)) \)[/tex]:
[tex]\[ g_A(f_A(x)) = g_A\left( \frac{x}{8} + 14 \right) = 8 \left( \frac{x}{8} + 14 \right) - 14 = x + 112 - 14 = x + 98 \neq x \][/tex]
Again, this does not simplify to [tex]\( x \)[/tex], confirming they are not inverses.
### Pair B
Functions:
[tex]\[ f_B(x) = 2x^3 - 11 \][/tex]
[tex]\[ g_B(x) = \frac{x + 11}{2} \][/tex]
1. Check [tex]\( f_B(g_B(x)) \)[/tex]:
[tex]\[ f_B(g_B(x)) = f_B\left( \frac{x + 11}{2} \right) = 2\left( \frac{x + 11}{2} \right)^3 - 11 \][/tex]
This is complex and a detailed check shows that it does not simplify to [tex]\( x \)[/tex].
2. Check [tex]\( g_B(f_B(x)) \)[/tex]:
[tex]\[ g_B(f_B(x)) = g_B(2x^3 - 11) = \frac{2x^3 - 11 + 11}{2} = \frac{2x^3}{2} = x^3 \neq x \][/tex]
Thus, they are not inverses.
### Pair C
Functions:
[tex]\[ f_C(x) = 3x^3 + 15 \][/tex]
[tex]\[ g_C(x) = \sqrt[3]{\frac{x}{3}} - 15 \][/tex]
1. Check [tex]\( f_C(g_C(x)) \)[/tex]:
[tex]\[ f_C(g_C(x)) = f_C\left( \sqrt[3]{\frac{x}{3}} - 15 \right) = 3\left( \sqrt[3]{\frac{x}{3}} - 15 \right)^3 + 15 \][/tex]
Again, a detailed check is needed, and it does not simplify to [tex]\( x \)[/tex].
2. Check [tex]\( g_C(f_C(x)) \)[/tex]:
[tex]\[ g_C(f_C(x)) = g_C(3x^3 + 15) = \sqrt[3]{\frac{3x^3 + 15}{3}} - 15 = \sqrt[3]{x^3 + 5} - 15 \neq x \][/tex]
So, they are not inverses.
### Pair D
Functions:
[tex]\[ f_D(x) = \frac{3}{x} - 6 \][/tex]
[tex]\[ g_D(x) = \frac{3}{x + 6} \][/tex]
1. Check [tex]\( f_D(g_D(x)) \)[/tex]:
[tex]\[ f_D(g_D(x)) = f_D\left( \frac{3}{x + 6} \right) = \frac{3}{\frac{3}{x + 6}} - 6 = x + 6 - 6 = x \][/tex]
This simplifies to [tex]\( x \)[/tex], satisfying the condition [tex]\( f_D(g_D(x)) = x \)[/tex].
2. Check [tex]\( g_D(f_D(x)) \)[/tex]:
[tex]\[ g_D(f_D(x)) = g_D\left( \frac{3}{x} - 6 \right) = \frac{3}{\left( \frac{3}{x} \right) - 6} = \frac{3x}{3 - 6x} = \frac{3x}{3 - 6x} = x \][/tex]
This also simplifies to [tex]\( x \)[/tex], satisfying the condition [tex]\( g_D(f_D(x)) = x \)[/tex].
Thus, the functions in Pair D are indeed inverses of each other.
Answer: The pair of functions that are inverses of each other is Pair D.
