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Question: Find the inverse of \(f(x) = \dfrac{{3x + 4}}{5}\)?...

Find the inverse of f(x)=3x+45f(x) = \dfrac{{3x + 4}}{5}?

Explanation

Solution

According to given in the question we have to determine the inverse of the function f(x)=3x+45f(x) = \dfrac{{3x + 4}}{5}. So, first of all to find the inverse of the given function we have to let the function have the same variable which can be (y, a, n, or, m).
Now, to find the inverse of the function f(x)f(x) which is f−1(x){f^{ - 1}}(x) we have to switch all xsxs and ysys.
Substitute the value of f−1(x){f^{ - 1}}(x) which we already let as any variable and then we have to solve the expression after substituting the value of f−1(x){f^{ - 1}}(x) after switching the variable xx as yy.
Then apply the cross-multiplication and multiply, add, subtract and divide the terms which can be multiplied, added, subtracted and divided.

Complete step by step solution:
First of all to find the inverse of the given function we have to let the function have the same variable which can be (y, a, n, or, m) which is as mentioned in the solution hint. Hence,
We have,
⇒y=3x+45\Rightarrow y = \dfrac{{3x + 4}}{5}
Now, to find the inverse of the function f(x)f(x) which is f−1(x){f^{ - 1}}(x) we have to switch all xsxs and ysys. Which is as mentioned in the solution hint.

Substitute the value of f−1(x){f^{ - 1}}(x) which we already let as any variable and then we have to solve the expression after substituting the value of f−1(x){f^{ - 1}}(x) after switching the variable xx as yy. Hence,
⇒x=3y+45\Rightarrow x = \dfrac{{3y + 4}}{5}

Now, apply the cross-multiplication and multiply, add, subtract and divide the terms which can be multiplied, added, subtracted and divided. Hence,
⇒5x=5(3y+4)5 ⇒5x=3y+4  \Rightarrow 5x = \dfrac{{5(3y + 4)}}{5} \\\ \Rightarrow 5x = 3y + 4 \\\
Now, we have to add and subtract 4 in the both sides of the expression which is as obtained just above,
⇒5x−4=3y+4−4 ⇒5x−4=3y  \Rightarrow 5x - 4 = 3y + 4 - 4 \\\ \Rightarrow 5x - 4 = 3y \\\
Now, we have to divide with the integer 3 in the both sides of the expression as obtained just above, hence,
⇒5x−43=3y3 ⇒5x−43=y  \Rightarrow \dfrac{{5x - 4}}{3} = \dfrac{{3y}}{3} \\\ \Rightarrow \dfrac{{5x - 4}}{3} = y \\\

Hence, we have determined the inverse off(x)=3x+45f(x) = \dfrac{{3x + 4}}{5}which is⇒5x−43=y \Rightarrow \dfrac{{5x - 4}}{3} = y.

Note:
• To determine the inverse of the function it is necessary that we have to switch the variable xx with the variable we have let in the starting of the solution which is yy.
• To substitute the value of f−1(x){f^{ - 1}}(x) which we already let as any variable and then we have to solve the expression after substituting the value of f−1(x){f^{ - 1}}(x) after switching the variable xx as yy.