Answer :
Sure, let's approach this step-by-step.
Given the expression:
[tex]\[ -\frac{1}{(4t-7)^2} \][/tex]
### Step 1: Identify the Inverse
The inverse of a function \( f(x) \) is denoted as \( f^{-1}(x) \) and is the function that "reverses" the effect of \( f(x) \).
If we have an expression \( -\frac{1}{(4t-7)^2} \), we are looking to find \( f^{-1}(x) \).
However, here we are asked to use inverse notation on the mathematical operation itself. Hence, we need to treat \(-\frac{1}{(4t-7)^2}\) as a function and find its multiplicative inverse.
### Step 2: Express the Inverse
The multiplicative inverse (or simply the inverse in the context of division) of a function \( f(t) \) is given by \( \frac{1}{f(t)} \).
Therefore, for the function:
[tex]\[ f(t) = -\frac{1}{(4t-7)^2} \][/tex]
The inverse function (multiplicative inverse) would be:
[tex]\[ \frac{1}{-\frac{1}{(4t-7)^2}} = - (4t-7)^2 \][/tex]
### Step 3: Simplify the Inverse Expression
After taking the inverse, we simplify the expression if possible.
[tex]\[ -\left(4t-7\right)^2 \][/tex]
This is already in its simplest form.
### Conclusion
Hence, the inverse notation of the given expression \(-\frac{1}{(4t-7)^2}\) is simply:
[tex]\[ -(4t-7)^2 \][/tex]
Given the expression:
[tex]\[ -\frac{1}{(4t-7)^2} \][/tex]
### Step 1: Identify the Inverse
The inverse of a function \( f(x) \) is denoted as \( f^{-1}(x) \) and is the function that "reverses" the effect of \( f(x) \).
If we have an expression \( -\frac{1}{(4t-7)^2} \), we are looking to find \( f^{-1}(x) \).
However, here we are asked to use inverse notation on the mathematical operation itself. Hence, we need to treat \(-\frac{1}{(4t-7)^2}\) as a function and find its multiplicative inverse.
### Step 2: Express the Inverse
The multiplicative inverse (or simply the inverse in the context of division) of a function \( f(t) \) is given by \( \frac{1}{f(t)} \).
Therefore, for the function:
[tex]\[ f(t) = -\frac{1}{(4t-7)^2} \][/tex]
The inverse function (multiplicative inverse) would be:
[tex]\[ \frac{1}{-\frac{1}{(4t-7)^2}} = - (4t-7)^2 \][/tex]
### Step 3: Simplify the Inverse Expression
After taking the inverse, we simplify the expression if possible.
[tex]\[ -\left(4t-7\right)^2 \][/tex]
This is already in its simplest form.
### Conclusion
Hence, the inverse notation of the given expression \(-\frac{1}{(4t-7)^2}\) is simply:
[tex]\[ -(4t-7)^2 \][/tex]
