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Sagot :
Any relation that has an inverse is a function
- The value of [tex]\mathbf{f^{-1}(33)}[/tex] is [tex]\mathbf{f^{-1}(33) = 67.5}[/tex]
- The rule of [tex]\mathbf{f^{-1}(n)}[/tex] is [tex]\mathbf{f^{-1}(n) = 2.5(n -6)}[/tex]
The function is given as:
[tex]\mathbf{f(n) = 0.4n + 6}[/tex]
(a) Evaluate [tex]\mathbf{f^{-1}(33)}[/tex]
First, we calculate the inverse function
We have:
[tex]\mathbf{f(n) = 0.4n + 6}[/tex]
Rewrite as:
[tex]\mathbf{y = 0.4n + 6}[/tex]
Subtract 6 from both sides
[tex]\mathbf{y -6= 0.4n + 6 - 6}[/tex]
[tex]\mathbf{y -6= 0.4n}[/tex]
Divide both sides by 0.4
[tex]\mathbf{\frac{1}{0.4}(y -6)= \frac{0.4n}{0.4}}[/tex]
[tex]\mathbf{\frac{1}{0.4}(y -6)= n}[/tex]
[tex]\mathbf{2.5(y -6)= n}[/tex]
Make n the subject
[tex]\mathbf{n = 2.5(y -6)}[/tex]
Rewrite as:
[tex]\mathbf{n = 2.5(f(n) -6)}[/tex]
So, the inverse function is:
[tex]\mathbf{f^{-1}(n) = 2.5(n -6)}[/tex]
Substitute 33 for n to calculate [tex]\mathbf{f^{-1}(33)}[/tex]
[tex]\mathbf{f^{-1}(33) = 2.5(33 -6)}[/tex]
[tex]\mathbf{f^{-1}(33) = 2.5(27)}[/tex]
[tex]\mathbf{f^{-1}(33) = 67.5}[/tex]
(b) The rule of [tex]\mathbf{f^{-1}(n)}[/tex]
In (a), we have: [tex]\mathbf{f^{-1}(n) = 2.5(n -6)}[/tex]
Hence, the rule of [tex]\mathbf{f^{-1}(n)}[/tex] is [tex]\mathbf{f^{-1}(n) = 2.5(n -6)}[/tex]
Read more about functions and inverses at:
https://brainly.com/question/10300045
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