Answer :
To determine which pairs of functions are inverses of each other, we need to check if they satisfy the condition \( f(g(x)) = x \) and \( g(f(x)) = x \) for each pair. Let's go through each pair step-by-step:
### Pair A
- \( f(x) = 7x^3 + 10 \)
- \( g(x) = \sqrt[3]{\frac{x - 10}{7}} \)
To check if these are inverses:
1. Compute \( f(g(x)) = f(\sqrt[3]{\frac{x - 10}{7}}) \)
2. Compute \( g(f(x)) = g(7x^3 + 10) \)
### Pair B
- \( f(x) = 2x^3 + 12 \)
- \( g(x) = \frac{x}{2} - 12 \)
To check if these are inverses:
1. Compute \( f(g(x)) = f\left(\frac{x}{2} - 12\right) \)
2. Compute \( g(f(x)) = g(2x^3 + 12) \)
### Pair C
- \( f(x) = \sqrt[3]{x + 2} - 7 \)
- \( g(x) = (x + 2)^3 + 7 \)
To check if these are inverses:
1. Compute \( f(g(x)) = f((x + 2)^3 + 7) \)
2. Compute \( g(f(x)) = g(\sqrt[3]{x + 2} - 7) \)
### Pair D
- \( f(x) = \frac{\sqrt[3]{x + 8}}{7} \)
- \( g(x) = (7x - 8)^3 \)
To check if these are inverses:
1. Compute \( f(g(x)) = f((7x - 8)^3) \)
2. Compute \( g(f(x)) = g\left(\frac{\sqrt[3]{x + 8}}{7}\right) \)
After evaluating the expressions for each pair, we determine if both conditions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true for each pair.
### Conclusion
Upon computation:
- Pair A, C, and D do not satisfy the inverse condition for both compositions.
- Pair B also does not satisfy the inverse condition.
Therefore, none of the given pairs of functions are inverses of each other. The answer is that none of the pairs are inverses, denoted by -1.
### Pair A
- \( f(x) = 7x^3 + 10 \)
- \( g(x) = \sqrt[3]{\frac{x - 10}{7}} \)
To check if these are inverses:
1. Compute \( f(g(x)) = f(\sqrt[3]{\frac{x - 10}{7}}) \)
2. Compute \( g(f(x)) = g(7x^3 + 10) \)
### Pair B
- \( f(x) = 2x^3 + 12 \)
- \( g(x) = \frac{x}{2} - 12 \)
To check if these are inverses:
1. Compute \( f(g(x)) = f\left(\frac{x}{2} - 12\right) \)
2. Compute \( g(f(x)) = g(2x^3 + 12) \)
### Pair C
- \( f(x) = \sqrt[3]{x + 2} - 7 \)
- \( g(x) = (x + 2)^3 + 7 \)
To check if these are inverses:
1. Compute \( f(g(x)) = f((x + 2)^3 + 7) \)
2. Compute \( g(f(x)) = g(\sqrt[3]{x + 2} - 7) \)
### Pair D
- \( f(x) = \frac{\sqrt[3]{x + 8}}{7} \)
- \( g(x) = (7x - 8)^3 \)
To check if these are inverses:
1. Compute \( f(g(x)) = f((7x - 8)^3) \)
2. Compute \( g(f(x)) = g\left(\frac{\sqrt[3]{x + 8}}{7}\right) \)
After evaluating the expressions for each pair, we determine if both conditions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true for each pair.
### Conclusion
Upon computation:
- Pair A, C, and D do not satisfy the inverse condition for both compositions.
- Pair B also does not satisfy the inverse condition.
Therefore, none of the given pairs of functions are inverses of each other. The answer is that none of the pairs are inverses, denoted by -1.
