- Define the function \(m(t) = -\frac{28}{5t}\).
- Verify the additive inverse condition: \(m(t) + p(t) = 0\). Since \(-\frac{28}{5t} + \frac{28}{5t} = 0\), \(p(t)\) is correct.
- Verify the multiplicative inverse condition: \(m(t) \cdot r(t) = 1\). Since \(\left(-\frac{28}{5t}\right) \cdot \left(-\frac{28}{5t}\right) = \frac{784}{25t^2}
eq 1\), \(r(t)\) is incorrect.
- Conclude that Angie is correct about the additive inverse only. \(\boxed{\text{Angie is correct about the additive inverse only.}}\)
Explanation
- Understanding Definitions of Inverses
We are given the function \(m(t) = -\frac{28}{5t}\). We need to determine if the proposed additive inverse \(p(t) = \frac{28}{5t}\) and the proposed multiplicative inverse \(r(t) = -\frac{28}{5t}\) are correct.
The additive inverse \(a(t)\) of a function \(m(t)\) is defined such that \(m(t) + a(t) = 0\).
The multiplicative inverse \(m'(t)\) of a function \(m(t)\) is defined such that \(m(t) \cdot m'(t) = 1\).
- Checking the Additive Inverse
To check if \(p(t) = \frac{28}{5t}\) is the additive inverse of \(m(t) = -\frac{28}{5t}\), we calculate their sum:
\[m(t) + p(t) = \left(-\frac{28}{5t}\right) + \left(\frac{28}{5t}\right) = 0.\]
Since the sum is 0, Angie is correct about the additive inverse.
- Checking the Multiplicative Inverse
To check if \(r(t) = -\frac{28}{5t}\) is the multiplicative inverse of \(m(t) = -\frac{28}{5t}\), we calculate their product:
\[m(t) \cdot r(t) = \left(-\frac{28}{5t}\right) \cdot \left(-\frac{28}{5t}\right) = \left(\frac{28}{5t}\right)^2 = \frac{784}{25t^2}.\]
Since this product is not equal to 1 (for all \(t\)), \(r(t)\) is not the multiplicative inverse. The correct multiplicative inverse would be \(m'(t) = \frac{1}{m(t)} = \frac{1}{-28/(5t)} = -\frac{5t}{28}\). Thus, Angie is incorrect about the multiplicative inverse.
- Conclusion
Based on our analysis, Angie is correct about the additive inverse only. The additive inverse \(p(t) = \frac{28}{5t}\) satisfies \(m(t) + p(t) = 0\), while the multiplicative inverse \(r(t) = -\frac{28}{5t}\) does not satisfy \(m(t) \cdot r(t) = 1\). Therefore, the correct conclusion is that Angie is correct about the additive inverse only. The final answer is \(\boxed{\text{Angie is correct about the additive inverse only.}}\)
Examples
In economics, additive and multiplicative inverses are essential concepts. An additive inverse represents a debt that perfectly offsets a specific amount of assets, leading to a net worth of zero. For example, if you have an asset valued at \(v\), the debt \(-v\) acts as its additive inverse. A multiplicative inverse, on the other hand, is crucial in currency conversion or rate adjustment; if a factor \(f\) is applied to a value to scale it, the multiplicative inverse \(1/f\) reverses that scaling effect exactly. Understanding these inverses helps in balancing budgets and normalizing financial data.