Certainly! Let’s analyze the function [tex]\( m(t) = \frac{-28}{5t} \)[/tex] and see if Angie is correct about the additive inverse and the multiplicative inverse.
### Additive Inverse
The additive inverse of a function [tex]\( m(t) \)[/tex] is a function that, when added to [tex]\( m(t) \)[/tex], results in zero. Mathematically, if [tex]\( m(t) \)[/tex] is [tex]\( f(t) \)[/tex], then the additive inverse would be [tex]\( -f(t) \)[/tex].
For [tex]\( m(t) = \frac{-28}{5t} \)[/tex], the additive inverse would be:
[tex]\[ - \left( \frac{-28}{5t} \right) = \frac{28}{5t} \][/tex]
Angie claims that the additive inverse is [tex]\( p(t) = \frac{28}{5t} \)[/tex], which is correct.
### Multiplicative Inverse
The multiplicative inverse of a function [tex]\( m(t) \)[/tex] is a function that, when multiplied by [tex]\( m(t) \)[/tex], results in one. Mathematically, if [tex]\( m(t) \)[/tex] is [tex]\( f(t) \)[/tex], then the multiplicative inverse would be [tex]\( \frac{1}{f(t)} \)[/tex].
For [tex]\( m(t) = \frac{-28}{5t} \)[/tex], the multiplicative inverse would be:
[tex]\[ \frac{1}{\frac{-28}{5t}} = \frac{5t}{-28} = \frac{-5t}{28} \][/tex]
Angie claims that the multiplicative inverse is [tex]\( r(t) = \frac{-28}{5t} \)[/tex], which is actually the original function [tex]\( m(t) \)[/tex] itself, not its multiplicative inverse.
### Conclusion
Angie is correct about the additive inverse only. The multiplicative inverse she provided is incorrect. Therefore, the correct response is:
Angie is correct about the additive inverse only.
This concludes our detailed analysis.
