Answer :
To determine which pair of functions are inverses of each other, we need to verify if \( f(g(x)) = x \) and \( g(f(x)) = x \) for each set of functions.
### Pair A:
\( f(x)=\frac{x}{7}+10 \) and \( g(x)=7x-10 \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f(7x - 10) = \frac{7x - 10}{7} + 10 = x - \frac{10}{7} + 10 = x - \frac{10}{7} + \frac{70}{7} = x + \frac{60}{7} \][/tex]
Since \( f(g(x)) \neq x \), these are not inverses.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{x}{7} + 10\right) = 7\left(\frac{x}{7} + 10\right) - 10 = x + 70 - 10 = x + 60 \][/tex]
Since \( g(f(x)) \neq x \), these are not inverses.
### Pair B:
\( f(x)=\frac{7}{x}-2 \) and \( g(x)=\frac{x+2}{7} \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x+2}{7}\right) = \frac{7}{\frac{x+2}{7}} - 2 = 7 \cdot \frac{7}{x+2} - 2 = \frac{49}{x+2} - 2 \][/tex]
Since \( f(g(x)) \neq x \), these are not inverses.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{7}{x}-2\right) = \frac{\frac{7}{x} - 2 + 2}{7} = \frac{7/x}{7} = \frac{1}{x} \][/tex]
Since \( g(f(x)) \neq x \), these are not inverses.
### Pair C:
\( f(x)=\sqrt[3]{11x} \) and \( g(x)=\left(\frac{x}{11}\right)^3 \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\left(\frac{x}{11}\right)^3\right) = \sqrt[3]{11 \left(\frac{x}{11}\right)^3} = \sqrt[3]{\frac{11x^3}{1331}} = \sqrt[3]{\frac{x^3}{1}} = x \][/tex]
This part holds true, but we need to verify the second condition.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\sqrt[3]{11x}\right) = \left(\frac{\sqrt[3]{11x}}{11}\right)^3 = \left(\frac{x}{11}\right)^3 \][/tex]
Since \( g(f(x)) \neq x \), these are not inverses.
### Pair D:
\( f(x)=9x-6 \) and \( g(x)=\frac{x+6}{9} \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x+6}{9}\right) = 9 \left(\frac{x+6}{9}\right) - 6 = x + 6 - 6 = x \][/tex]
This holds true.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g(9x-6) = \frac{9x-6 + 6}{9} = \frac{9x}{9} = x \][/tex]
This also holds true.
Since both conditions for inverses are satisfied in pair D, \( f(x)=9x-6 \) and \( g(x)=\frac{x+6}{9} \), we conclude that the pair of functions in D are inverses of each other.
Therefore, the correct answer is:
[tex]\[ D. f(x) = 9x - 6 \text{ and } g(x) = \frac{x+6}{9} \][/tex]
### Pair A:
\( f(x)=\frac{x}{7}+10 \) and \( g(x)=7x-10 \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f(7x - 10) = \frac{7x - 10}{7} + 10 = x - \frac{10}{7} + 10 = x - \frac{10}{7} + \frac{70}{7} = x + \frac{60}{7} \][/tex]
Since \( f(g(x)) \neq x \), these are not inverses.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{x}{7} + 10\right) = 7\left(\frac{x}{7} + 10\right) - 10 = x + 70 - 10 = x + 60 \][/tex]
Since \( g(f(x)) \neq x \), these are not inverses.
### Pair B:
\( f(x)=\frac{7}{x}-2 \) and \( g(x)=\frac{x+2}{7} \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x+2}{7}\right) = \frac{7}{\frac{x+2}{7}} - 2 = 7 \cdot \frac{7}{x+2} - 2 = \frac{49}{x+2} - 2 \][/tex]
Since \( f(g(x)) \neq x \), these are not inverses.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\frac{7}{x}-2\right) = \frac{\frac{7}{x} - 2 + 2}{7} = \frac{7/x}{7} = \frac{1}{x} \][/tex]
Since \( g(f(x)) \neq x \), these are not inverses.
### Pair C:
\( f(x)=\sqrt[3]{11x} \) and \( g(x)=\left(\frac{x}{11}\right)^3 \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\left(\frac{x}{11}\right)^3\right) = \sqrt[3]{11 \left(\frac{x}{11}\right)^3} = \sqrt[3]{\frac{11x^3}{1331}} = \sqrt[3]{\frac{x^3}{1}} = x \][/tex]
This part holds true, but we need to verify the second condition.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g\left(\sqrt[3]{11x}\right) = \left(\frac{\sqrt[3]{11x}}{11}\right)^3 = \left(\frac{x}{11}\right)^3 \][/tex]
Since \( g(f(x)) \neq x \), these are not inverses.
### Pair D:
\( f(x)=9x-6 \) and \( g(x)=\frac{x+6}{9} \)
1. Compute \( f(g(x)) \):
[tex]\[ f(g(x)) = f\left(\frac{x+6}{9}\right) = 9 \left(\frac{x+6}{9}\right) - 6 = x + 6 - 6 = x \][/tex]
This holds true.
2. Compute \( g(f(x)) \):
[tex]\[ g(f(x)) = g(9x-6) = \frac{9x-6 + 6}{9} = \frac{9x}{9} = x \][/tex]
This also holds true.
Since both conditions for inverses are satisfied in pair D, \( f(x)=9x-6 \) and \( g(x)=\frac{x+6}{9} \), we conclude that the pair of functions in D are inverses of each other.
Therefore, the correct answer is:
[tex]\[ D. f(x) = 9x - 6 \text{ and } g(x) = \frac{x+6}{9} \][/tex]
